The differential equation of y=e^x(a+bx+x^2) | vedclass

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(C) Given $y=e^x(a+bx+x^2)$.First,differentiate with respect to $x$:$\frac{dy}{dx} = e^x(a+bx+x^2) + e^x(b+2x)$$\frac{dy}{dx} = y + e^x(b+2x) \quad ..

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

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(C) Given $y=e^x(a+bx+x^2)$.First,differentiate with respect to $x$:$\frac{dy}{dx} = e^x(a+bx+x^2) + e^x(b+2x)$$\frac{dy}{dx} = y + e^x(b+2x) \quad ...(i)$Now,differentiate again with respect to $x$:$\frac{d^2 y}{dx^2} = \frac{dy}{dx} + e^x(b+2x) + e^x(2)$Substitute $e^x(b+2x) = \frac{dy}{dx} - y$ from equation $(i)$:$\frac{d^2 y}{dx^2} = \frac{dy}{dx} + (\frac{dy}{dx} - y) + 2e^x$$\frac{d^2 y}{dx^2} = 2\frac{dy}{dx} - y + 2e^x$Rearranging the terms,we get:$\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} - 2e^x + y = 0$.

The differential equation of $y=e^x(a+bx+x^2)$ is

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