Chapter 5.7, Problem 24E

(a)

To determine

To calculate: The particular solution of the provided differential equation $f^{\prime }\left(x\right)=x^{\frac{2}{5}}+x$ when the initial condition is, when $f\left(1\right)=-7$.

(b)

To determine

To prove: The particular solution, $f\left(x\right)=\frac{5}{7}{x}^{\frac{7}{5}}+\frac{1}{2}{x}^2-\frac{115}{14}$ obtained in part (a) is correct by substituting into the differential equation $f^{\prime }\left(x\right)=x^{\frac{2}{5}}+x$.