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gogolik [260]
3 years ago
9

Use the method for solving equations of the form StartFraction dy Over dx EndFraction equals Upper G (ax plus by )to solve the f

ollowing 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.)
Mathematics
1 answer:
g100num [7]3 years ago
5 0

\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}

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