Q. 8 - 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 $f^{\text{'}}\left(x\right)=$ $f\left(x\right), f\left(0\right)=3$.

Step-by-Step Solution

Answer

Ans: The initial-value problem $f^{\text{'}}\left(x\right)=f\left(x\right),f\left(0\right)=3$.

Step 1. Given information:

The initial-value problem $f^{\text{'}}\left(x\right)=f\left(x\right),f\left(0\right)=3$.

Step 2. Solving the differential equation :

In order to solve the given differential equation $f^{\text{'}}\left(x\right)=f\left(x\right)$, observe that the solution will be a function whose derivative is the function itself. Observe that such a function is the exponential function $e^x$ because the derivative is $\frac{d}{dx}\left(e^x\right)=e^x$.

So, one such function that satisfies the given differential equation could be $f\left(x\right)=C{e}^x$.

Now use the initial condition $f\left(0\right)=3$ and simplify further for C to get

$$\begin{gathered} f\left(0\right)=C{e}^0 \\ 3=C\left(1\right) \\ 3=C \\ \Rightarrow C=3 \end{gathered}$$

Thus, the solution of the given differential equation is $f\left(x\right)=3e^x$.