Find the general solution for the following differential equation $\frac{{{\mathrm{d}}^{2}}y}{\mathr

Question

Find the general solution for the following differential equation $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}={{\mathrm{e}}^{-5x+3}}+\sin x$. [3] By using the substitution $z=x+\frac{\mathrm{d}y}{\mathrm{d}x}$, show that the following differential equation $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-\frac{\mathrm{d}y}{\mathrm{d}x}-x+1=0$ can be reduced to $\frac{\mathrm{d}z}{\mathrm{d}x}=z$. [2] Hence, given that when $x=0$, $y=1$ and $\frac{\mathrm{d}y}{\mathrm{d}x}=1$, find $y$ in terms of $x$. [4]

Accepted Answer

(a) $y = \frac{1}{25}e^{-5x+3} - \sin x + Cx + D$ (b) $y = e^x - \frac{x^2}{2}$

2019 MI P2 Q3

2019 MI P2 Q3

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