Question

Use the method for solving equations of the form StartFraction dy Over dx EndFraction equals Upper G (ax plus by )to solve the following differential equation. StartFraction dy Over dx EndFraction equals 2 sine (4 x minus 2 y )Ignoring lost solutions, if any, an implicit solution in the form F(x,y)=C is ___ = C (Type an expression using x and y as the variables.)

Answer 1

$\dfrac{\mathrm dy}{\mathrm dx}=2\sin(4x-2y)$

Not sure what method you're referring to... but one substitution comes to mind: let $v(x)=4x-2y(x)$, so that $\frac{\mathrm dv}{\mathrm dx}=4-2\frac{\mathrm dy}{\mathrm dx}$. Then

$\dfrac12\left(4-\dfrac{\mathrm dv}{\mathrm dx}\right)=2\sin v\implies\dfrac{\mathrm dv}{\mathrm dx}=4(1-\sin v)$

This ODE is separable, as

$\dfrac{\mathrm dv}{1-\sin v}=4\,\mathrm dx$

Integrate both sides; on the left, we have

$\dfrac1{1-\sin v}=\dfrac{1+\sin v}{1-\sin^2v}=\dfrac{1+\sin v}{\cos^2v}=\sec^2v+\sec v\tan v$

which has a recognizable antiderivative, giving us

$\tan v+\sec v=4x+C$

$\implies\tan(4x-2y)+\sec(4x-2y)=4x+C$

so that the solution is

$F(x,y)=\boxed{\tan(4x-2y)+\sec(4x-2y)-4x=C}$