# Solve the following $\sec 2x dy -\sin 5x \sec^{2} ydx=0$

Toolbox:
• First order , first degree DE
• Variable separable : Variables of a DE are rearranged to separate them, ie
• $f_1(x)g_2(y)dx+f_2(x)g_1(y)dy=0$
• Can be written as $\large\frac{g_1 (y)}{g_2(y)}$$dy=-\large\frac{f_1(x)}{f_2(x)}$$dx$
• The solution is therefore $\int \large\frac{g_1(y)}{g_2(y)}$$dy=-\int \large\frac{f_1(x)}{f_2(x)}$$dx+c$
Step 1:
$\sec 2x dy - \sin 5x \sec^2 y dx=0$ divided by $\sec 2x \sec^2 y$
$\large\frac{dy}{\sec^2 y}-\frac{\sin 5x}{\sec 2x}$$dx=0 \cos ^2 y dy= \sin 5x \cos 2x dx Step 2: Integrating \int \cos^2 y dy =\int \sin 5x \cos 2x dx +c_1 \int \large\frac{1+\cos 2y}{2}dy=\large\frac{1}{2}$$ \int (\sin \large\frac{7x}{2}$$+ \sin 3x)dx+c_1 \large\frac{y}{2}+\frac{1}{4}$$ \sin 2y = -\large\frac{1}{2} \times \frac{1}{7} $$\cos 7x-\large\frac{1}{2} \times \large\frac{1}{3}$$\cos 3x +c_1$
Multiply by two and rearrange