Question

The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).

Answer 1

According to the question,
$\frac{dy}{dx}=x^2$
$$\begin{gathered} \Rightarrow dy=x^{2}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ \int dy=\int x^{2}dx \\ \Rightarrow y=\frac{x^3}{3}+C \\ \mathrm{Since}\mathrm{the}\mathrm{curve}\mathrm{passes}\mathrm{through}\left(-1,1\right),\mathrm{it}\mathrm{satisfies}\mathrm{the}\mathrm{above}\mathrm{equation}. \\ \therefore 1=\frac{-1}{3}+C \\ \Rightarrow C=1+\frac{1}{3} \\ \Rightarrow C=\frac{4}{3} \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}C,\mathrm{we}\mathrm{get} \\ y=\frac{x^3}{3}+\frac{4}{3} \\ \Rightarrow 3y=x^3+4 \end{gathered}$$