Solved Exercises on Differential Equations

A collection of worked-out exercises involving first- and second-order differential equations.

First-Order Differential Equations

Exercise	$$ y' + 1 = 0 $$
Exercise	$$ y' + y = 0 $$
Exercise	$$ y' + y = x $$
Exercise	$$ y' + 2x = 0 $$
Exercise	$$ y' + y = 1 $$
Exercise	$$ xy' + y = 0 $$
Exercise	$$ xy' - y = 0 $$
Exercise	$$ y' + xy = 0 $$
Exercise	$$ y' + xy + x = 0 $$
Exercise	$$ y' + 2xy = 0 $$
Exercise	$$ y' + y \cdot \sin x = 0 $$
Exercise	$$ xy' + 2y = x $$
Exercise	$$ 2yy' - x^2 = 0 $$
Exercise	$$ 3y^2 \cdot y' + x = 0 $$
Exercise	$$ y' + y \cos x = \cos x $$
Exercise	$$ 2x (y^2 + 1) - y' = 0 $$
Exercise	$$ xyy' = 2x^2 + y^2 $$
Exercise	$$ 2xyy' = x^2 + 2y^2 $$
Exercise	$$ y' - x \cdot \cos x = 0 $$
Exercise	$$ x^2 y' = xy + y^2 $$
Exercise	$$ 3y^2 y' = x + 1 $$
Exercise	$$ yy' - 3x^2 = 2 $$
Exercise	$$ y' = - \frac{1 + y}{x^2} $$
Exercise	$$ y' + \frac{y}{x} = \frac{\sin x}{x} $$
Exercise	$$ \begin{cases} y' = y - 1 \\ \\ y(0) = 7 \end{cases} $$
Exercise	$$ \begin{cases} y' = \cos^2 y \\ \\ y(0) = 0 \end{cases} $$
Exercise	$$ \begin{cases} y' - x^2 = 0 \\ x = 4 \\ y = \frac{1}{4} \end{cases} $$
Exercise	$$ \begin{cases} y' = y^2 \cdot x^3 \\ \\ y(0) = 7 \end{cases} $$
Exercise	$$ \begin{cases} y' = y^2 \cdot t^3 \\ \\ y(0) = 0 \end{cases} $$
Exercise	$$ \begin{cases} y' = \frac{-t}{y} \\ \\ y(0) = 5 \end{cases} $$