Question

if y=e^2x (ax+b), then prove that d^2 y/dx^2 -4 dy/dx+4y=0

Answer 1

$\textbf{Given:}$

$y=e^{2x}(a\,x+b)$

$\textbf{To prove:}$

$\dfrac{d^2y}{dx^2}-4\,\dfrac{dy}{dx}+4\,y=0$

$\textbf{Solution:}$

$\text{Consider,}$

$y=e^{2x}(a\,x+b)$

$\text{This can be written as}$

$y\,e^{-2x}=a\,x+b$

$\text{Differentiate with respect to 'x'}$

$y\,e^{-2x}(-2)+e^{-2x}\,\dfrac{dy}{dx}=a$

$e^{-2x}(-2y+\dfrac{dy}{dx})=a$

$\text{Differentiate again with respect to 'x'}$

$e^{-2x}(-2\dfrac{dy}{dx}+\dfrac{d^2y}{dx^2})+(-2y+\dfrac{dy}{dx})e^{-2x}(-2)=0$

$e^{-2x}[-2\dfrac{dy}{dx}+\dfrac{d^2y}{dx^2}+4y-2\dfrac{dy}{dx}]=0$

$e^{-2x}[\dfrac{d^2y}{dx^2}-4\dfrac{dy}{dx}+4y]=0$

$\text{But}\;e^{-2x}{\neq}0$

$\implies\boxed{\bf\,\dfrac{d^2y}{dx^2}-4\dfrac{dy}{dx}+4y=0}$

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