Q. 2 - Calculus

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.

Step-by-Step Solution

Answer

Ans:  $f\left(x\right)=-\cos x+C$

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 $f\left(x\right)=-\cos x+C$.