Question

$$\frac{dy}{dx}=x^5+x^2-\frac{2}{x},x\ne 0$$

Answer 1

We have,
$$\frac{dy}{dx}=x^5+x^2-\frac{2}{x}$$
$$\Rightarrow dy=(x^5+x^2-\frac{2}{x})dx$$
Integrating both sides, we get
$$\Rightarrow \int dy=\int (x^5+x^2-\frac{2}{x})dx$$
$$\Rightarrow y=\frac{x^6}{6}+\frac{x^3}{3}-2\log \left|x\right|+C$$
$$\text{Clearly, }y=\frac{x^6}{6}+\frac{x^3}{3}-2\log \left|x\right|+\text{ C is defined for all }x\in R\text{ except }x=0.$$
$$\text{ Hence, }y=\frac{x^6}{6}+\frac{x^3}{3}-2\log \left|x\right|+\text{ C, where }x\in R-\left\{0\right\},\text{ is the solution to the given differential equation.}$$