Solve the following differential equations using the Product Rule of Differentiation \[\frac{\mathrm

Question

Solve the following differential equations using the Product Rule of Differentiation

\[\frac{\mathrm{d}}{\mathrm{d}x}(\mathrm{f}(y)\mathrm{g}(x)) = \mathrm{g}(x)\mathrm{f}'(y)\frac{\mathrm{d}y}{\mathrm{d}x} + \mathrm{f}(y)\mathrm{g}'(x)\]

$x\frac{\mathrm{d}y}{\mathrm{d}x} + y = e^x$
$\cos x \frac{\mathrm{d}y}{\mathrm{d}x} - y \sin x = x^2$
$\frac{1}{x} \frac{\mathrm{d}y}{\mathrm{d}x} - \frac{y}{x^2} = \sin x$

Accepted Answer

(a) $xy=e^x+C$, so $y=\frac{e^x+C}{x}$ for $x\ne0$; $C=-1$ permits the extension $y(0)=1$. (b) $y\cos x=\frac{x^3}{3}+C$, so $y=\frac{\frac{x^3}{3}+C}{\cos x}$ where $\cos x\ne0$, with removable extensions as stated in the working. (c) $y=Cx-x\cos x$, $x\ne0$.

First order linear equation

First order linear equation

Solution:

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