# Does this equation fall in any of the linear, homogeneous, exact categories?

Solve: $$\frac{dy}{dx}-\frac{\tan y}{1+x}=(1+x)e^x\sec y$$

It's homework. I've been taught linear, homogeneous, and exact differential equations. I don't think this is linear or homogeneous. I tried exact. I multiplied both sides by $(1+x)dx$, then got all the terms on the left hand side to get it in the form $Mdx+Ndy=0$.

Turns out that the partial derivative condition for exact equations is not satisfied. So, I tried to calculate an integrating factor. I tried both x dependent and y dependent factors. But they both came out to be dependent on both variables.

Does this problem really fall in any of the categories: linear, homogeneous, exact? If it doesn't, what kind of problem is this?

• Please use MathJax to facilitate research. Commented May 7, 2018 at 13:40

## 1 Answer

It is neither a linear nor a homogeneous differential equation.
However, multiplying both sides of the equation by $\cos(y)$, we get $$\cos(y) \frac{d y}{d x} - \frac{sin(y)}{1 +x} = (1 +x) \exp(x)$$
And making the substition $Y = \sin(y)$, the equation becomes $$\frac{d Y}{d x} -\frac{Y}{1 +x} = (1 +x) \exp(x)$$ which is a linear differential equation of order 1.