IGNOU MTE 8 2018 - Using the method of undetermined coefficients, find the general soluti

Question

Using the method of undetermined coefficients, find the general solution of the DE y^iv - 2y^m + 2yⁿ. = 3e^-x + 2e^-x x + e^-xsinx.

13 Feb 2018

Answer :

Word Count : 370

The given differential equation is: \[ y^{(4)} - 2y^{(m)} + 2y^{(n)} = 3e^{-x} + 2e^{-x}x + e^{-x}\sin(x) \] We can solve this using the method of undetermined coefficients. However, we need to clarify a few things: 1. The expression contains terms like \(y^{(m)}\) and \(y^{(n)}\). To proceed, I'll assume that the correct form of the equation is: \[ y^{(4)} - 2y'' + 2y' = 3e^{-x} + 2xe^{-x} + e^{-x}\sin(x) \] If this assumption is correct, we can solve this as follows: ### Step 1: Solve the homogeneous equation The homogeneous part of the equation is: \[ y^{(4)} - 2y'' + 2y' = 0 \] We assume a solution of the form \(y_h = e^{rx}\). Substituting into the homogeneous equation gives: \[ r^4 e^{rx} - 2r^2 e^{rx} + 2r e^{rx} = 0 \] Factoring out \(e^{rx}\) (which is never zero): \[ r^4 - 2r^2 + 2r = 0 \]

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