Help to find asymptotic solution of linear ode: KSy_1 = (y_1)"(x) w/ BC (dy_1/dx)_{x=

[flanker]

Help to find asymptotic solution of linear ode: KSy_1 = (y_1)"(x) w/ BC (dy_1/dx)_{x=

I have a trouble with ODE, I try to find asymptotic solution for odes which presented in pics. But I can’t. Please introduce a method which I solve these equations. I can solve these equations analytically but after solution, inverse Laplace transform must apply to find final answer. In analytical solution inverse Laplace transform near to impossible. Since I want a way to approximate analytical solution to these equations afterwards I can apply inverse Laplace transform to find final answer.
thanks

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\(\displaystyle \large{ KSy_1(x)\, =\, y_1^{"}(x) }\). . . . . . . . . . . . . . . . .\(\displaystyle \large{ BC:\, \left(\dfrac{dy_1}{dx}\right)_{x = -1}\, =\, \dfrac{A}{s},\, \left(\dfrac{dy_1}{dx}\right)_{x = 1}\, =\, 0 }\)

\(\displaystyle \large{ KSy_2(x)\, -\, y_2^{"}(x)\, +\, y_1^{'}(x)\, =\, 0 }\). . . . .\(\displaystyle \large{ BC:\, \left(\dfrac{dy_2}{dx}\right)_{x = -1}\, =\, 0,\, \mbox{ }\, \left(\dfrac{dy_2}{dx}\right)_{x = 1}\, =\, 0 }\)

\(\displaystyle \large{ A,\, K\, \mbox{ and }\, S\, \mbox{ are fixed values.} }\)

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[Deleted member 4993]

I have a trouble with ODE, I try to find asymptotic solution for odes which presented in pics. But I can’t. Please introduce a method which I solve these equations. I can solve these equations analytically but after solution, inverse Laplace transform must apply to find final answer. In analytical solution inverse Laplace transform near to impossible. Since I want a way to approximate analytical solution to these equations afterwards I can apply inverse Laplace transform to find final answer.
thanks

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\(\displaystyle \large{ KSy_1(x)\, =\, y_1^{"}(x) }\). . . . . . . . . . . . . . . . .\(\displaystyle \large{ BC:\, \left(\dfrac{dy_1}{dx}\right)_{x = -1}\, =\, \dfrac{A}{s},\, \left(\dfrac{dy_1}{dx}\right)_{x = 1}\, =\, 0 }\)

\(\displaystyle \large{ KSy_2(x)\, -\, y_2^{"}(x)\, +\, y_1^{'}(x)\, =\, 0 }\). . . . .\(\displaystyle \large{ BC:\, \left(\dfrac{dy_2}{dx}\right)_{x = -1}\, =\, 0,\, \mbox{ }\, \left(\dfrac{dy_2}{dx}\right)_{x = 1}\, =\, 0 }\)

\(\displaystyle \large{ A,\, K\, \mbox{ and }\, S\, \mbox{ are fixed values.} }\)

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[Ishuda]

I have a trouble with ODE, I try to find asymptotic solution for odes which presented in pics. But I can’t. Please introduce a method which I solve these equations. I can solve these equations analytically but after solution, inverse Laplace transform must apply to find final answer. In analytical solution inverse Laplace transform near to impossible. Since I want a way to approximate analytical solution to these equations afterwards I can apply inverse Laplace transform to find final answer.
thanks

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\(\displaystyle \large{ KSy_1(x)\, =\, y_1^{"}(x) }\). . . . . . . . . . . . . . . . .\(\displaystyle \large{ BC:\, \left(\dfrac{dy_1}{dx}\right)_{x = -1}\, =\, \dfrac{A}{s},\, \left(\dfrac{dy_1}{dx}\right)_{x = 1}\, =\, 0 }\)

\(\displaystyle \large{ KSy_2(x)\, -\, y_2^{"}(x)\, +\, y_1^{'}(x)\, =\, 0 }\). . . . .\(\displaystyle \large{ BC:\, \left(\dfrac{dy_2}{dx}\right)_{x = -1}\, =\, 0,\, \mbox{ }\, \left(\dfrac{dy_2}{dx}\right)_{x = 1}\, =\, 0 }\)

\(\displaystyle \large{ A,\, K\, \mbox{ and }\, S\, \mbox{ are fixed values.} }\)

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You say you can solve these equations analytically. What is the solution?

 

[flanker]

You say you can solve these equations analytically. What is the solution?

yes analytical solution for these equations is available. But it is important to applied inverse laplace transform to answer of these equations to find final answer. it is notable that these equations are created with application of laplace transform to a PDE. Thus i have to solve these equations and apply inverse laplace transform to treate the PDE answer. please help me to find a esay way to solve my problem by using software like mathematica thanks