The equation is.
$\cos(x) \cdot y'+\sin(x) \cdot y = 2(\cos(x))^3 \cdot \sin(x)-1$
a) Find all the solutions of the associate homogeneous equation. Let $S_h$(homogeneous) denote such a set of solutions.
b) Find all the solutions of the given equation. Let $S_{nh}$(non homogeneous) denote such a set of solutions.
c) Explain in detail what is the relationship between $S_h$ and $S_nh$.
d) Find the specific solution $y \in S_{nh}$ that satisfies $y(\frac{\pi}{4}) = 3 \cdot \sqrt(2)$
The solution of the equation is below :$$\cos (x)\cdot y'+\sin (x)\cdot y=2(\cos (x))^{ 3 }\cdot \sin (x)-1\\ \cos (x)\cdot y'+\sin (x)\cdot y=0\\ \cos (x)\cdot y'=-\sin (x)\cdot y\\ \int { \frac { dy }{ y } } =-\int { \frac { \sin { x } }{ \cos { x } } dx } \\ \ln { y } =\ln { C\cos { x } } \\ y=C\cos { x } \\ y=C\left( x \right) \cos { x } \\ { y }^{ \prime }={ C }^{ \prime }\left( x \right) \cos { x } -C\left( x \right) \sin { x } \\ { C }^{ \prime }\left( x \right) \cos ^{ 2 }{ x } -C\left( x \right) \sin { x } \cos { x } +C\left( x \right) \cos { x } \sin { y } =2(\cos (x))^{ 3 }\cdot \sin (x)-1\\ { C }^{ \prime }\left( x \right) \cos ^{ 2 }{ x } =2(\cos (x))^{ 3 }\cdot \sin (x)-1\\ C\left( x \right) =\int { \left( 2\cos { x\sin { x } -\frac { 1 }{ \cos ^{ 2 }{ x } } } \right) dx } =\sin ^{ 2 }{ x } -\tan { x } +C$$