Sagot :
Given:
[tex]\frac{dy}{dx}=x^{\frac{3}{2}}[/tex]To find the general solution:
It can be written as,
[tex]dy=x^{\frac{3}{2}}dx[/tex]Taking integration on both sides we get,
[tex]\begin{gathered} \int dy=\int x^{\frac{3}{2}}dx \\ y=\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+c \\ y=\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+c \\ y=\frac{2}{5}x^{\frac{5}{2}}+c \end{gathered}[/tex]Hence, the general solution is,
[tex]y=\frac{2}{5}x^{\frac{5}{2}}+c[/tex]To check the result by differentiation:
Differentiating with respect to x we get,
[tex]\begin{gathered} \frac{dy}{dx}=\frac{2}{5}(\frac{5}{2})x^{\frac{5.}{2}-1}+0 \\ \frac{dy}{dx}=x^{\frac{3}{2}} \end{gathered}[/tex]Hence, it is verified.