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) $y = \frac{1}{x}(e^x + c)$ ; (b) $y = \sec x \left( \frac{1}{3} x^3 + c \right)$ ; (c) $y = cx - x \cos x$.

First order linear equation

First order linear equation