Question

In problems 23–28 Find an explicit solution to the given initial-value problem.

$\frac{\text{dx}}{\text{dt}}\text{= 4}\left({\text{x}}^{\text{2}}\text{+ 1}\right)\text{,}\text{x (}\pi \text{/4) = 1}$

Short answer

$x=\tan \left(4t-\frac{3\pi }{4}\right)$

Step-by-step solution

Definition of separable equation

A first-order differential equation of the form$\frac{dy}{dx}=g\left(x\right)h\left(y\right)$ is said to be separable or to have separable variables.

Separate the variables and integration

Separate the variables.

$\frac{dx}{x^2+1}=4dt$

Integrate both sides.

$\int \frac{dx}{x^2+1}=\int 4dt$

Do the integration.

$$\begin{gathered} {\tan }^{-1}x=4t+C \\ x=\tan (4t+C)······\left(1\right) \end{gathered}$$

Substitute the initial condition

Substitute the initial condition. $x(\pi /4)=1$

$$\begin{gathered} 1=\tan \left(4\right(\pi /4)+C) \\ 1=\tan (\pi +C) \\ {\tan }^{-1}\left(1\right)=\pi +C \\ \frac{\pi }{4}=\pi +C \\ C=-\frac{3\pi }{4} \end{gathered}$$

Substitute the C value in (1),

$x=\tan \left(4t-\frac{3\pi }{4}\right)$

Therefore, the solution is $\text{x = tan}\left(\text{4t -}\frac{\text{3}\pi }{\text{4}}\right)$