Q. 7 - Calculus

Question

Initial-Value Problems: An initial-value problem is a differential equation together with an initial condition that specifies one value of a function.

• Find the solution $f$ to the initial-value problem $f^{\text{'}}\left(x\right)=\sin \left(e^x\right)$   $,f\left(\pi \right)=0$.

Step-by-Step Solution

Answer

Ans:  The solution to the given initial value problem is $f\left(x\right)=Si\left(e^x\right)-Si\left(e^{\pi }\right)$

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

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

Apply initial condition $f\left(\pi \right)=0$

$$\begin{gathered} f\left(\pi \right)=Si\left(e^{\pi }\right)+C \\ 0=Si\left(e^{\pi }\right)+C \\ C=-Si\left(e^{\pi }\right) \\ \Rightarrow f\left(x\right)=Si\left(e^x\right)-Si\left(e^{\pi }\right) \end{gathered}$$

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

Hence, the solution of the given differential equation is $f\left(x\right)=Si\left(e^x\right)-Si\left(e^{\pi }\right)$