In problem 7-16, solve the equation.$\frac{\mathrm{dx}}{\mathrm{dt}}\mathbf{=}\mathbf{3}{\mathbf{xt}}^2$

A first-order ordinary differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}\mathbf{f}\left(x,y\right)$is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions g(x)that is a function of x alone and h(y) that is a function of y alone.

A separable differential equation can be expressed as $\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}\mathbf{g}\left(x\right)\mathbf{h}\left(y\right)$. By separating the variables, the equation follows $\frac{\mathrm{dy}}{\mathbf{h}\left(y\right)}\mathbf{=}\mathbf{g}\left(x\right)\mathbf{dx}$. Then, on direct integration of both sides, the solution of the differential equation is determined.

The given equation is

$\frac{\mathrm{dx}}{\mathrm{dt}}=3\mathrm{x}{\mathrm{t}}^2\cdot \cdot \cdot \cdot \cdot \cdot \left(1\right)$

After separating the variables, equation (1) can be written as

$\frac{\mathrm{dx}}{\mathrm{x}}=3{\mathrm{t}}^2\mathrm{dt}\cdot \cdot \cdot \cdot \cdot \cdot \left(2\right)$

Integrate both sides of equation (2). It results,

$\begin{array}{c}\int \frac{\mathrm{dx}}{\mathrm{x}}=\int 3{\mathrm{t}}^2\mathrm{dt}\\ \ln \mathrm{x}=3\left[\frac{{\mathrm{t}}^3}{3}\right]+\ln \mathrm{C}\left[\ln \mathrm{C}=\mathrm{IntegratingConstant}\right]\\ \ln \mathrm{x}-\ln \mathrm{C}={\mathrm{t}}^3\\ \ln \left(\frac{\mathrm{x}}{\mathrm{C}}\right)={\mathrm{t}}^3\left[\ln \mathrm{a}-\ln \mathrm{b}=\ln \left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right]\end{array}$

$\begin{array}{c}\frac{\mathrm{x}}{\mathrm{C}}={\mathrm{e}}^{{\mathrm{t}}^3}\\ \mathrm{x}=\mathrm{C}{\mathrm{e}}^{{\mathrm{t}}^3}\end{array}$

Therefore, the solution of the given equation is$\mathbf{x}\left(t\right)\mathbf{=}\mathbf{C}{\mathbf{e}}^{{\mathbf{t}}^3}$.