Use a substitution to turn $\cos(y)y'+ \tan(x)\sin(y)=\sin(x)$ into a linear differential equation.

Question

$$\cos(y)\frac{dy}{dx} + \tan(x)\sin(y)=\sin(x)$$ Is there any method to find a substitution to fit this purpose? Or is it purely done through brute force/trial and error?

Answer 1

You can rewrite the equation this way..

$$\cos(y)\frac{dy}{dx} + \tan(x)\sin(y)=\sin(x)$$ $$ \implies (\sin(y))' + \tan(x)\sin(y)=\sin(x)$$ And the equation becomes linear of first order $$ \implies z' + \tan(x)z=\sin(x)$$ Where $z=\sin y$

Answer 2

Hint: $$\int{\cos(y)\frac{dy}{dx}dx}=\int{\cos(y)dy}=\sin y$$