Question

Differential Equations: A differential equation is simply an equation that involves derivatives. The solution of a differential equation is the family of functions that make the differential equation true.

• Solve the differential equation $f^{\text{'}}\left(x\right)=\sin x$. Remember, the solution will be a family of functions.

Short answer

Ans: $f\left(x\right)=-\cos x+C$

Step-by-step solution

Step 1. Given information: 

The differential equation, $f^{\text{'}}\left(x\right)=\sin x$.

Step 2. Solving the differential equation :

In order to solve the given differential equation $f^{\text{'}}\left(x\right)=\sin x$,

Since the differential equation is already separable, so integrate both sides with respect to $\mathrm{x}$to get

$$\begin{gathered} \int f^{\text{'}}\left(x\right)=\int \sin xdx \\ f\left(x\right)=-\cos x+C \end{gathered}$$

Here C is the integration constant.

Thus, the solution of the given differential equation is$f\left(x\right)=-\cos x+C$.