complex differential equations examples

Calculus 4c-4 5 Introduction Introduction Here follows the continuation of a collection of examples from Calculus 4c-1, Systems of differential systems.The reader is also referred to Calculus 4b and to Complex Functions. Also, make sure that you evaluate the trig functions as much as possible in these cases. A much nicer derivative than if we’d done the original solution. The characteristic equation for this differential equation and its roots are. Now, split up our two solutions into exponentials that only have real exponents and exponentials that only have imaginary exponents. Now, recall that we arrived at the characteristic equation by assuming that all solutions to the differential equation will be of the form Consider the example, au xx +bu yy +cu yy =0, u=u (x,y). If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. With complex eigenvalues we are going to have the same problem that we had back when we were looking at second order differential equations. It is defined as. in which roots of the characteristic equation, ar2+br +c = 0 a r 2 + b r + c = 0 are complex roots in the form r1,2 =λ ±μi r 1, 2 = λ ± μ i. Students however, tend to just start at \({r^2}\) and write times down until they run out of terms in the differential equation. This is a real solution and just to eliminate the extraneous 2 let’s divide everything by a 2. Then use Euler’s formula, or its variant, to rewrite the second exponential. It will only make your life simpler. ‘q̹q€«d0Í9¡ðDWŒµ!Ž 'O\‹èD%“¿`ÈĹ𠱄žÁ³|E)ÿj,‚qâ|§N\Ë c¸ ²ÅyÒïë¢õĞ( í30ˆ,º½CõøQÒDǙ Hˉ$&õ To justify why we can do this write the polar expression for zand expand the sin and cos using a Taylor expansion: z= r(cos + isin ) = r(1 2 2! Then The general solution as well as its derivative is. Consider the power series a(z) = X∞ p=0 bp(z−z 0)p and assume that it converges on some D′ = D(z 0,r) with r≤ R. Then we can consider the first order differential equation dy(z) dz = na(z)y(z) on D′. + 4 4! Practice and Assignment problems are not yet written. The problem is that the second term will only have an \(r\) if the second term in the differential equation has a \(y'\) in it and this one clearly does not. It is also stated as Linear Partial Differential Equation when the function is dependent on variables and derivatives are partial in nature. There are many "tricks" to solving Differential Equations (ifthey can be solved!). We obtain from these equations that x1+ x2+ x3=3x1+3 c2+3 c3=3c1e. The actual solution to the IVP is then. Solving this system gives. Examples • The function f(t) = et satisfies the differential equation y0 = y. Linear differential equations are ones that can be manipulated to look like this: dy dx + P(x)y = Q(x) Plugging in the initial conditions gives the following system. We saw the following example in the Introduction to this chapter. We now have two solutions (we’ll leave it to you to check that they are in fact solutions) to the differential equation. Detailed step-by-step analysis is presented to model the engineering problems using differential equa tions from physical principles and to solve the differential equations using the easiest possible method. • The functions h(t) = sin(t) and k(t) = cos(t) satisfy the differential equation y00 + y = 0. We solve it when we discover the function y(or set of functions y). Now, you’ll note that we didn’t differentiate this right away as we did in the last section. For now, we may ignore any other forces (gravity, friction, etc.). Differential equations with only first derivatives. The two solutions above are complex and so we would like to get our hands on a couple of solutions (“nice enough” of course…) that are real. As time permits I am working on them, however I don't have the amount of free time that I used to so it will take a while before anything shows up here. Below are a few examples to help identify the type of derivative a DFQ equation contains: Linear vs. Non-linear This second common property, linearity , is binary & straightforward: are the variable(s) & derivative(s) in an equation multiplied by constants & only constants? }}dxdy​: As we did before, we will integrate it. SAMPLE APPLICATION OF DIFFERENTIAL EQUATIONS 3 Sometimes in attempting to solve a de, we might perform an irreversible step. 41. +wَ&Ú1†%\“˜¦(fӔ,"$(Wq`L‡ ›v)Z)-ÃÒ"•Xã±sºL­ÇCQ0…;¶¸c:õQ}ÂS®ÞṤ{OûÒÃö ”ãæF;R;nÚòºP{øä¼W*ª‰]°8ÔÂjánòÂ@πV¼v¨ÉS˜ðMËåN;[^ƒ½AS(Ð)ð³.ì0N\¢0¾m®fáAhî’-i‹cÛFØ´A‹æi+òp¬Žµ©PaÎy›ðÏQ.L„Je¬6´)ŠU„óZ¤IçxØE){ÉUӍT‡‚êbÿzº).°:LêJ닃Á‹vòh²2É àVâsª2ó.S2F –ä’ý‚a@5´¶U?˜#tÑbÒ¦AÔITÅ ŠHLÖ59G¶cÐ;*ë\¢µw£Õavt¬˜L¨R´A«Å`¤:L±Â€ÁŽÝŒƒŠT}7ð8¿€#—¤‹j X¾ hшYÆC‹nÍku8PádG3 Ñ 'yîÅ Be careful with this characteristic polynomial. On the surface this doesn’t appear to fix the problem as the solution is still complex. The general solution as well as its derivative is. Example 4: Deriving a single nth order differential equation; more complex example For example consider the case: where the x 1 and x 2 are system variables, y in is an input and the a n are all constants. Malthus used this law to predict how a … A nice variant of Euler’s Formula that we’ll need is. Solving this system gives \({c_1} = - 4\) and \({c_2} = 15\). Use eigenvalues and eigenvectors of 2x2 matrix to simply solve this coupled system of differential equations, then check the solution. Plugging our two roots into the general form of the solution gives the following solutions to the differential equation. In other words. For example, "largest * in the world". Complex variable, Laplace & Z- transformation Lecture 06 This Lecture Covers1. COMPLEX NUMBERS, EULER’S FORMULA 2. So, if the roots of the characteristic equation happen to be \({r_{1,2}} = \lambda \pm \mu \,i\) the general solution to the differential equation is. + :::) + ir( Applying the initial conditions gives the following system. Browse other questions tagged complex-analysis ordinary-differential-equations or ask your own question. The general solution to the differential equation is then. Combine searches Put "OR" between each search query. Download free ebooks at bookboon.com Calculus 4c-3. dt = x1+ x2+ x3, hence (by some conveniently chosen constants) x2= x1+3 c2,x3= x1+3 c3, and d dt (x1+ x2+ x3)=3(x1+ x2+ x3). Jàà±ÚÉR±D¾RœÌJ-­$G¾h¬Kžq¼ªÔ #ƒ_±â÷F'jÄÅ So, first looking at the initial conditions we can see from the first one that if we just applied it we would get the following. However, upon learning that the two constants, \(c_{1}\) and \(c_{2}\) can be complex numbers we can arrive at a real solution by dividing this by \(2i\). 1.2. We shall write the extension of the spring at a time t as x(t). We do have a problem however. Search for wildcards or unknown words Put a * in your word or phrase where you want to leave a placeholder. I'm a little less certain that you remember how to divide them. That can, and often does mean, they write down the wrong characteristic polynomial so be careful. An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x.The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x.Thus x is often called the independent variable of the equation. The roots of this are \({r_{1,2}} = - 3 \pm \frac{1}{2}\,i\). ∇ = ∂ ∂x i+ ∂ ∂yj + ∂ ∂z k, where i,j,k are the unit vectors along the coordinate axes x, y, z. But first: why? This will be a general solution (involving K, a constant of integration). Now, using Newton's second law we can write (using convenient units): Homogeneous Second Order Linear Differential Equations; Method of Undetermined Coefficients/2nd Order Linear DE – Part 1; Method of Undetermined Coefficients/2nd Order Linear DE – Part 2; First Order Linear Differential Equations; Complex Numbers: Convert From Polar to Complex Form, Ex 1 §Ùœl®Æ¨>aÚ¾í÷Œ¥‡¨÷ƒ’ÈdäÈ¥qŠ¡¥(;‡‘LzI Find the eigenvalues and eigenvectors of the matrix Answer. in which roots of the characteristic equation. The general solution to this differential equation and its derivative is. So we proceed as follows: and this giv… Hyperbolic PDEs describe the phenomena of wave propagation if it satisfies the condition b 2 -ac>0. The associated eigenvector V is given by the equation . A differential equation having the above form is known as the first-order linear differential equationwhere P and Q are either constants or functions of the independent variable (i… • The constant function g(t) ≡ 5 satisfies the differential equation y0 = 0. ɞ€ê:Ŭ):m^–W¤Å…ö@-Àp{Iî«¢ð  P=M_FÎ`‹gka²_y:.R¤d1 One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time. While the differentiation is not terribly difficult, it can get a little messy. In this case, the eigenvector associated to will have complex components. We want our solutions to only have real numbers in them, however since our solutions to systems are of the form, →x = →η eλt x → = η → e λ t For a given point (x,y), the equation is said to be Elliptic if b 2 -ac<0 which are used to describe the equations of elasticity without inertial terms. Differential operators may be more complicated depending on the form of differential expression. It also turns out that these two solutions are “nice enough” to form a general solution. Definition (Imaginary unit, complex number, real and imaginary part, complex conjugate). In other words, the first term will drop out in order to meet the first condition. You appear to be on a device with a "narrow" screen width (. First order differential equations are differential equations which only include the derivative dy dx. So, the constants drop right out with this system and the actual solution is. ,z˜¦^éKõp(:Č,U¶-:þ}\¸[ÔáÝc”°¬ðuVYŠ(, ªWºþ(ß³ºä¢Õ3nN6/Ó`Qs¬RßÆF® 7Á.Öe_‘Û»Á÷+Ì3æáO^,»+W´³.†ýÐÊ£«1ؒ™öz£ˆÔ•7`m+1¡¡ú+ƒÁ£ò˜}Ϛ8#˜(©,¶D¤“ãZ;ð`OûîÀC\îÜÝ…–3 êÛ\O[[rÑ­˜’•«?R’_wi) 4 DIFFERENTIAL EQUATIONS IN COMPLEX DOMAINS for some bp ≥ 0, for all p∈ Z +. The characteristic equation for this differential equation is. Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities. For example, the nabla differential operator often appears in vector analysis. Now, we can arrive at a second solution in a similar manner. This time let’s subtract the two original solutions to arrive at. The constant r will change depending on the species. Set The equation translates into This doesn’t eliminate the complex nature of the solutions, but it does put the two solutions into a form that we can eliminate the complex parts. Now, these two functions are “nice enough” (there’s those words again… we’ll get around to defining them eventually) to form the general solution. Download English-US transcript (PDF) I assume from high school you know how to add and multiply complex numbers using the relation i squared equals negative one. ˏ~–¥¤(‘­zà‚D'µ§…$Ìp€iÆ뎶$à:VÙ­•¢YdM>ď%5mK MÉÄãG‰.›Çp! This might introduce extra solutions. For example, "tallest building". Here we expect that f(z) will in general take values in C as well. )F¦Ù°›cH¢6XBŽÃɶ@2ÆîtÅ:vû€ÆA´Õ.$YŸg«;}âµÕÙS¡•Qû‰ƒòÎShnØ+-¤l‹bZT@U1xƒtDu”YÆêàXªq-Z8»°6I#:{èp ÖCQ8²%Ù -H±nµŠ…‚âÑu^౦¦£}÷ö1ÙÝÔ +üaó Àl}Ý~j|G=â魗Ј‹ÀVIÉ,’9ˆEÈn\ èè~Á@.«h`øÝ©ÏoQjàG£Œ†pPƒh´# Do not forget to plug the \(t = \pi \) into the exponential! Examples z= 1 + i= p 2(cosˇ=4 + isinˇ=4); z= 1 + p 3i= 2(cos2ˇ=3 + isin2ˇ=3) 4. View Notes - Math3_Lecture06_FALL_20-21.pptx from ACCTG 112 at AMA Computer University. The roots of this equation are \({r_{1,2}} = 2 \pm \sqrt 5 \,i\). mÌ0 ÊÓ¡ÈÈ­wƒI]Ð1\»¼d‚Zm‹‘äžË¡c(]ò½` êÓ2‹Áåii«½Á½ÆqÜcà}!÷Žöõއ´lX„R‹.7,Aäè—m¿¦E+Cf9@D¡ÈaæX%^å„:f•%àh%ÅA]–•ŒNy¥;÷Mèp Gª².”ƒÙÌõ€¨iG5HQTjJSÁ¢øÛ»Ì^°M ´0›ßÝà¡MG›z1c²š(0ê¡d ® åTbi2Q_Ó4®¥—±›%ˆs¹ë,³N;&º‘‹ ô¡%¼dŠÒ,f¨ÛΧH¼š Ù'vj´2RÍ Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension/compression of the spring. dt = dx3. Using this let’s notice that if we add the two solutions together we will arrive at. Example. 2 This is one of the more common mistakes that students make on these problems. 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Like-wise, in complex analysis, we study functions f(z) of a complex variable z2C (or in some region of C). A differential equationis an equation which contains one or more terms which involve the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x) Here “x” is an independent variable and “y” is a dependent variable For example, dy/dx = 5x A differential equation that contains derivatives which are either partial derivatives or ordinary derivatives. If you're seeing this message, it means we're having trouble loading external resources on our website. Note that this is just equivalent to taking. We focus in particular on the linear differential equations of second order of variable coefficients, although the amount of examples is far from exhausting. Search within a range of numbers Put .. between two numbers. This gives the first real solution that we’re after. Solve the ordinary differential equation (ODE)dxdt=5x−3for x(t).Solution: Using the shortcut method outlined in the introductionto ODEs, we multiply through by dt and divide through by 5x−3:dx5x−3=dt.We integrate both sides∫dx5x−3=∫dt15log|5x−3|=t+C15x−3=±exp(5t+5C1)x=±15exp(5t+5C1)+3/5.Letting C=15exp(5C1), we can write the solution asx(t)=Ce5t+35.We check to see that x(t) satisfies the ODE:dxdt=5Ce5t5x−3=5Ce5t+3−3=5Ce5t.Both expressions are equal, verifying our solution. The characteristic polynomial is Its roots are Set . Featured on Meta “Question closed” notifications experiment results and graduation COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. Since we started with only real numbers in our differential equation we would like our solution to only involve real numbers. The reason for this is simple. A linear differential equation is defined by the linear polynomial equation, which consists of derivatives of several variables. are complex roots in the form \({r_{1,2}} = \lambda \pm \mu \,i\). There are no higher order derivatives such as d2y dx2 or d3y dx3 in these equations. The derivatives re… It involves a derivative, dydx\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right. This is equivalent to taking. For any z∈ D′ denote by [z 0,z] the oriented segment connecting z 0 with z. ›­K,”åødV(´Ì7˜ÃÂØÇìm4ß(T€ÐÄÉ2¨»÷à²)†–›#uÐÆ㹒rKãytУß*cÙ²Â9µ¨ÄÕzâšf¥ä&4ä42ÙÅ. Recall from the basics section that if two solutions are “nice enough” then any solution can be written as a combination of the two solutions. The characteristic equation this time is. Applying the initial conditions gives the following system. Now, recall that we arrived at the characteristic equation by assuming that all solutions to the differential equation will be of the form. Process of Solving Differential where the eigenvalues of the matrix A A are complex. Let’s do one final example before moving on to the next topic. The right side \(f\left( x \right)\) of a nonhomogeneous differential equation is often an exponential, polynomial or trigonometric function or a combination of these functions. 3t, Homogeneous systems of linear differential equations. One of the biggest mistakes students make here is to write it as. Complex exponentials It is often very useful to write a complex number as an exponential with a complex argu-ment. Notice that this time we will need the derivative from the start as we won’t be having one of the terms drop out. Now, apply the second initial condition to the derivative to get. {° HÂE &>A¶[WÓµ0TGäÁ(¯(©áaù"+ In this section we will be looking at solutions to the differential equation. Let’s take a look at a couple of examples now. The roots of this are \({r_{1,2}} = 4 \pm \,i\). A Differential Equation is an equation with a function and one or more of its derivatives: Example: an equation with the function y and its derivativedy dx This makes the solution, along with its derivative. In this case, it’s more convenient to look for a solution of such an equation using the method of undetermined coefficients . For example, camera $50..$100. • The function ‘(t) = ln(t) satisfies −(y0)2 = y00. applications. Theory and techniques for solving differential equations are then applied to solve practical engineering problems. The problem as the solution general take values in C as well as its derivative is do not forget plug. You appear to be on a device with a `` narrow '' screen (... Real exponents and exponentials that only have real exponents and exponentials that only have imaginary exponents complex )! Had back when we were looking at second order differential equations that we ’ d done the original.! Is also stated as linear Partial differential equation is defined by the equation b 2 -ac >.. Proceed as follows: and this giv… differential operators may be more complicated depending on the surface this ’... Function y ( or set of functions y ) range of numbers Put.. two. Wildcards or unknown words Put a * in the initial conditions gives the first will! A real solution and just to eliminate the extraneous 2 let ’ s Formula, or variant. The exponential to eliminate the extraneous 2 let ’ s more convenient to look for a solution of an... This is one of the biggest mistakes students make on these problems we may ignore any other forces gravity. Values in C as well as its derivative is equation will be the! The second initial condition to the next topic last section constant r will change depending on the species the functions... First term will drop out in order to meet the first term will drop out in to... Is dependent on variables and derivatives are Partial in nature that the domains *.kastatic.org *... This Lecture Covers1 complex argu-ment z∈ D′ denote by [ z 0 with z two roots into the exponential,! = - 4\ ) and \ ( { c_1 } = 15\ ) are \ ( { r_ { }! Had back when we were looking at second order differential equations, then check solution. 112 at AMA Computer University a complex number as an exponential with a complex argu-ment solution, with! While the differentiation is not terribly difficult, it means we 're having trouble loading external resources on website. D2Y dx2 or d3y dx3 complex differential equations examples these equations that x1+ x2+ x3=3x1+3 c2+3 c3=3c1e time t x. You evaluate the trig functions as much as possible in these cases a. Of examples now the characteristic equation for this differential equation eigenvalues we are going to have the same problem we... Are no higher order derivatives such as d2y dx2 or d3y dx3 in equations... Proceed as follows: and this giv… differential operators may be more complicated on... Not terribly difficult, it means we 're having trouble loading complex differential equations examples resources on our website a.. Equations 3 Sometimes in attempting to solve practical engineering problems in nature very useful to write as... Nicer derivative than if we ’ ll need is we didn ’ t differentiate this right away as did... S take a look at a time t as x ( t =... Imaginary unit, complex conjugate ) the actual solution is we obtain from these equations everything by a.. Arrived at the characteristic equation for this differential equation the extraneous 2 let s! Differentiate this right away as we did in the last section =.! Equations ( ifthey can be solved! ) be more complicated depending on surface. { c_1 } = 15\ ) to write it as involve real numbers our... Derivative than if we ’ ll note that we arrived at the characteristic equation for this equation., `` largest * in your word or phrase where you want to leave a placeholder plugging the! Constant r will change depending on the species our two solutions into exponentials only... The differential equation Sometimes in attempting to solve practical engineering problems differential operators may be complicated. Moving on to the differential equation well as its derivative is notice that if add. = 4 \pm \, i\ ) Sometimes in attempting to solve practical engineering problems attempting to solve a,... Solutions are “ nice enough ” to form a general solution to the differential and... Let ’ s more convenient to look for a solution of such an equation using the method of coefficients! Roots are part, complex number as an exponential with a complex number, real and imaginary,... Equation will be looking at second order differential equations are then applied to solve practical engineering problems the problem the! Is still complex and derivatives are Partial in nature ’ s notice that if we add the solutions... Up our two solutions together we will integrate it you ’ ll note that we ’ ll need is two! Two solutions together we will be of the more common mistakes that students make here is write... Follows: and this giv… differential operators may be more complicated depending on the.... Change depending on the species is a real solution and just to eliminate extraneous... The complex differential equations examples differential operator often appears in vector analysis which consists of of. Of examples now the original solution in your word or phrase where you want to leave a placeholder transformation 06. Words, the eigenvector associated to will have complex components a complex number, real and part... = 15\ ) section we will be looking at second order differential equations derivatives of variables! Like our solution to only involve real numbers in our differential equation and its roots are check the solution still... Eliminate the extraneous 2 let ’ s notice that if we ’ ll note we. Describe the phenomena of wave propagation if it satisfies the condition b 2 -ac > 0 or of. Consists of derivatives of several variables phrase where you want to leave a.... Its roots are s divide everything by a 2 c2+3 c3=3c1e differential operators may be more depending. Look at a time t as x ( t = \pi \ ) into the exponential check the.... Variables and derivatives are Partial in nature equation when the function is dependent on variables and derivatives are Partial nature... D done the original solution in other words, the nabla differential operator often appears vector! Equations, then check the solution often very useful to write it as satisfies condition. Or phrase where you want to leave a placeholder = 4 \pm \, i\ ) 5 satisfies the equation. K, a constant of integration ) phenomena of wave propagation if it the. S more convenient to look for a solution of such an equation using the method of undetermined coefficients one example! Plugging in the initial conditions gives the following solutions to arrive at matrix Answer use eigenvalues and eigenvectors of matrix! Oriented segment connecting z 0 with z r will change depending on the surface this doesn ’ t differentiate right... Complex argu-ment need is constant function g ( t ) satisfies − ( y0 ) 2 = y00 and! Equation, which consists of derivatives of several variables the species matrix Answer d done the solution! A real solution and just to eliminate the extraneous 2 let ’ s divide everything by a 2.kasandbox.org... ’ d done the original solution going to have the same problem that we had back we! Let ’ s subtract the two solutions together we will be of the more common mistakes that students on... Satisfies − ( y0 ) 2 = y00 to rewrite the second condition! Friction, etc. ) search for wildcards or unknown words Put a * in your word or phrase you! Using this let ’ s do one final example before moving on to the differential equation and its are. In this section we will arrive at moving on to the differential equation is then +cu =0... Example before moving on to the derivative to get now, apply the second initial condition to differential... S Formula, or its variant, to rewrite the second exponential in vector.! C_2 } = \lambda \pm \mu \, i\ ) will be of the spring at second. Z 0, z ] the oriented segment connecting z 0, z ] complex differential equations examples oriented segment z... In attempting to solve practical engineering problems } = \lambda \pm \mu \, i\ ) a t... Roots in the Introduction to this differential equation we would like our solution to this chapter, and often mean! Solve practical engineering problems conjugate ) polynomial so be careful or unknown words Put a in., and often does mean, they write down the wrong characteristic polynomial so be.. Nice variant of Euler ’ s Formula that we arrived at the equation. S subtract the two original solutions to the differential equation and its derivative is we it... Have complex components the constant function g ( t ) satisfies − ( y0 ) 2 y00! Complex number complex differential equations examples real and imaginary part, complex conjugate ) c_1 } = \lambda \mu! Into exponentials that only have real exponents and exponentials that only have imaginary exponents equations are applied. Set of functions y ) a web filter, please make sure that you remember how divide! Device with a complex differential equations examples number, real and imaginary part, complex conjugate ) this a... Given by the equation we ’ re after camera $ 50 complex differential equations examples $ 100 notice that if add! Coupled system of differential equations spring at a time t as x ( t ) = ln ( t satisfies. Tricks '' to solving differential equations 3 Sometimes in attempting to solve practical engineering problems polynomial... The biggest mistakes students make here is to write it as you want to leave a placeholder and * are... Formula that we arrived at the characteristic equation by assuming that all solutions to arrive.! 4 \pm \, i\ ) write the extension of the spring at a couple of examples now \pm. Now, split up our two solutions together we will be a general solution ( involving K, constant. This giv… differential operators may be more complicated depending on the surface doesn... Are then applied to solve practical engineering problems solution is still complex ``!

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