2. (a) Find the general solution of the equation $$ y^{\prime \prime}-4 y^{\prime}+4 y=(x+1) e^{2 x} $$ using the method of variation of parameters. [10 marks] (b) Determine the-eigenvalues and associated eigenfunctions for theendpoint problem $$ \cdots, \quad y(L)=0,(L0) . \quad $$ [10 marks]

Question

2. (a) Find the general solution of the equation $$ y^{\prime \prime}-4 y^{\prime}+4 y=(x+1) e^{2 x} $$ using the method of variation of parameters. [10 marks] (b) Determine the-eigenvalues and associated eigenfunctions for theendpoint problem $$ \cdots, \quad y(L)=0,(L>0) . \quad $$ [10 marks]

Lyons Weber · Accepted Answer

(a) The general solution of the differential equation is $$ y = C_1 e^{2x} + C_2 x e^{2x} + y_p $$. (b) The eigenvalues are $$ \lambda_n = \left(\frac{n\pi}{L}\right)^2 $$ and the associated eigenfunctions are $$ y_n(x) = B_n \sin\left(\frac{n\pi}{L} x\right) $$.

2. (a) Find the general solution of the equation \( y^{\prime \prime}-4 y^{\prime}+4 y=(x+1) e^{2 x} \) using the method of variation of parameters. [10 marks] (b) Determine the-eigenvalues and associated eigenfunctions for theendpoint problem \( \cdots, \quad y(L)=0,(L>0) . \quad \) [10 marks]

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2. (a) Find the general solution of the equation \( y^{\prime \prime}-4 y^{\prime}+4 y=(x+1) e^{2 x} \) using the method of variation of parameters. [10 marks] (b) Determine the-eigenvalues and associated eigenfunctions for theendpoint problem \( \cdots, \quad y(L)=0,(L>0) . \quad \) [10 marks]

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Respuesta rápida

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