Q. 3 - 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 $f^{\text{'}}\left(x\right)=\sin \left(e^x\right)$. The solution will be a family of nonelementary functions.

Step-by-Step Solution

Answer

Ans: The solution of the given differential equation is $f\left(x\right)=Si\left(e^x\right)+C$.

Step 1. Given information:

Consider the differential equation $f^{\text{'}}\left(x\right)=\sin \left(e^x\right)$

Step 2. Solving the differential equation :

Integrate on both sides

$$\begin{gathered} \int f^{\text{'}}\left(x\right)=\int \sin \left(e^x\right)dx \\ f\left(x\right)=\int \frac{\sin u}{u}du \left(\because u=\sin \left(e^x\right)\right) \\ f\left(x\right)=Si\left(u\right) \left(\because \int \frac{\sin u}{u}du=Si\left(u\right)\right) \\ f\left(x\right)=Si\left(e^x\right)+C \end{gathered}$$

Here, $Si\left(e^x\right)$ is the nonelementary function.

Hence, the solution of the given differential equation is $f\left(x\right)=Si\left(e^x\right)+C$