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Let CER, 20 € Domain(y), and k Range(y). Construct an example of a second order, ordinary differential equation and verify a solution using the following steps. (a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y"(x). (b) Choose functions (they may be constant) a2(x) and a(z). Then compute f(x) = a₂(x)y"(x) + a₁(x)y'(x) (c) Show that y(x) + C is a solution of f(x) = a₂(x)y"(x) + a₁(x)y'(x).

Let CER, 20 € Domain(y), and k Range(y). Construct an example of a second order, ordinary differential equation and verify a solution using the following steps. (a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y"(x). (b) Choose functions (they may be constant) a2(x) and a(z). Then compute f(x) = a₂(x)y"(x) + a₁(x)y'(x) (c) Show that y(x) + C is a solution of f(x) = a₂(x)y"(x) + a₁(x)y'(x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let CER, 20 € Domain(y), and k Range(y). Construct an example of a second order, ordinary differential equation and verify a solution using the following steps. (a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y"(x). (b) Choose functions (they may be constant) a2(x) and a(z). Then compute f(x) = a₂(x)y"(x) + a₁(x)y'(x) (c) Show that y(x) +C is a solution of f(x) = a₂(x)y"(x) + a₁(x)y'(x). (d) For what value(s) of C does y(zo) = k? Choose ro and k to give an example of an initial value problem using the ODE constructed in part (b). Provide the solution of the IVP. (e) Use a similar method (or share your own method) to construct another second-order IVP that you can verify the solution of without using any ODE technques other than the FTOC. Make sure to include justification for the solution and verify it.
Transcribed Image Text:Let CER, 20 € Domain(y), and k Range(y). Construct an example of a second order, ordinary differential equation and verify a solution using the following steps. (a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y"(x). (b) Choose functions (they may be constant) a2(x) and a(z). Then compute f(x) = a₂(x)y"(x) + a₁(x)y'(x) (c) Show that y(x) +C is a solution of f(x) = a₂(x)y"(x) + a₁(x)y'(x). (d) For what value(s) of C does y(zo) = k? Choose ro and k to give an example of an initial value problem using the ODE constructed in part (b). Provide the solution of the IVP. (e) Use a similar method (or share your own method) to construct another second-order IVP that you can verify the solution of without using any ODE technques other than the FTOC. Make sure to include justification for the solution and verify it.
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