Question

Solve the given differential equation

$\mathbf{(}\mathbf{2}{\mathbf{r}}^{\mathbf{2}}\mathbf{cos\theta sin\theta }\mathbf{+}\mathbf{rcos\theta }\mathbf{)}\mathbf{d\theta }\mathbf{+}\mathbf{(}\mathbf{4}\mathbf{r}\mathbf{+}\mathbf{sin\theta }\mathbf{-}\mathbf{2}{\mathbf{rcos}}^{\mathbf{2}}\mathbf{\theta }\mathbf{)}$dr = 0

Short answer

The general solution of the given differential equation is

${\mathrm{r}}^2\sin \left[2\right]\left(\mathrm{\theta }\right)+\mathrm{rsin}\left(\mathrm{\theta }\right)+{\mathrm{r}}^2+\mathrm{c}=0$

Step-by-step solution

We have functions

$$\begin{gathered} \underset{=M(\theta ,r)}{\underbrace{\left(2r^2\cos \left(\theta \right)\sin \left(\theta \right)+r\cos \left(\theta \right)\right)}}\mathrm{d}\theta +\underset{=N(\theta ,r)}{\underbrace{(4r+\sin (\theta )-2r\cos [2\left]\right(\theta \left)\right)}}\mathrm{d}r=0 \\ \begin{array}{l}\frac{\partial M}{\partial r}=4r\cos \left(\theta \right)\sin \left(\theta \right)+\cos \left(\theta \right)\\ \frac{\partial N}{\partial \theta }=\cos \left(\theta \right)+4r\cos \left(\theta \right)\sin \left(\theta \right)\end{array} \end{gathered}$$

This means there exists some function such that

$\begin{array}{l}\frac{\partial f}{\partial \theta }=M\\ \frac{\partial f}{\partial r}=N\end{array}$

Integrating the given functions