Problem 24 - Essential Calculus

Question

Solve the differential equation. $$ \frac{d y}{d x}=\frac{10 x^{2}}{\sqrt{1+x^{3}}} $$

Step-by-Step Solution

Answer

The general solution to the differential equation is \(2\sqrt{1+x^{3}} = \frac{10x^{3}}{3} + C\), where C is the constant of integration.

Step 1: Recognize the separable form

The given differential equation \(\frac{d y}{d x}=\frac{10 x^{2}}{\sqrt{1+x^{3}}}\) is already in the separable form \(g(x) * h(y)\). Here, \(g(x) = 10x^2\) and \(h(y) = \frac{1}{\sqrt{1+x^{3}}}\). So, the next step is to separate the variables y and x on either side of the equation.

Step 2: Separate the variables

We rewrite the equation in the separable form and move dx to the other side so that we have \(\frac{d y}{\sqrt{1+x^{3}}} = 10 x^{2} dx\). Now y and x are separated on either side, so each side can be integrated independently.

Step 3: Integrate both sides

Integrate both sides of the equation using the power rule for integration. Integration for \(\frac{dy}{\sqrt{1+x^{3}}}\) yields \(2\sqrt{1+x^{3}}+C_1\) (since the derivative of \(1+x^{3}\) is \(3x^2\)), and for \(10x^{2}dx\) yields \(\frac{10x^{3}}{3}+C_2\).

Step 4: Combine constants and simplify

Combine constants \(C_1\) and \(C_2\) into a single constant \(C = C_1 - C_2\), and simplify the equation to give the general solution. So, the general solution is \(2\sqrt{1+x^{3}} = \frac{10x^{3}}{3} + C\)