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