Question

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

• If $f_1\left(x\right)$ and $f_2\left(x\right)$ are both solutions of the differential equation $f^{\text{'}}\left(x\right)=g\left(x\right)$, where $g\left(x\right)$ is continuous, then what can you say about the relationship between $f_1$ and $f_2$ ? What could you say if, in addition, $f_1$ and $f_2$ agreed on any given value?

Short answer

Ans: $f_1$ and $f_2$ are linearly independent.

Step-by-step solution

Step 1. Given information:

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

Step 2. Solving the differential equation :

$f_1$and $f_2$are the solutions of the differential equation $f^{\text{'}}\left(x\right)=g\left(x\right)$Then,

$$\begin{gathered} f^{\text{'}}\left(x\right)=g\left(x\right) \\ f\left(x\right)=\int g\left(x\right)dx=f_1\left(x\right)+f_2\left(x\right) \end{gathered}$$

Thus, $f_1$ and $f_2$ are linearly independent.