Question

Solve the given differential equation by using an appropriate substitution. The DE is of the form dy/dx = f(Ax + By + C), which is given in (5) of Section 2.5. dy/dx = tan2(x + y)

Answer 1

$\dfrac{\mathrm dy}{\mathrm dx}=\tan^2(x+y)$

Let $v=x+y$, so that $\frac{\mathrm dv}{\mathrm dx}=1+\frac{\mathrm dy}{\mathrm dx}$:

$\dfrac{\mathrm dv}{\mathrm dx}-1=\tan^2v$

Recall that $\tan^2v+1=\sec^2v$:

$\dfrac{\mathrm dv}{\mathrm dx}=\sec^2v$

Separate the variables and integrate both sides:

$\cos^2v\,\mathrm dv=\mathrm dx$

$\dfrac v2+\dfrac{\sin(2v)}4=x+C$

Solve in terms of $y$:

$\dfrac{x+y}2+\dfrac{\sin(2x+2y)}4=x+C$