Show that if $y_1\left(x\right)and{y}_2\left(x\right)$ are both solutions of the differential equation $\frac{dy}{dx}=ky$ then the sum y1(x) + y2(x) is also a solution of the differential equation.

The given function is$y_1\left(x\right)and{y}_2\left(x\right)$

Recall that a function y(x) is defined as a solution of a differential equation if it makes the differential equation true, that is, it satisfies the differential equation. Since both $y_1\left(x\right)+y_2\left(x\right)$

are given to be solutions of the differential equation $\frac{dy}{dx}=ky$it Implies that both functions must

satisfy the differential equation. Therefore,

role="math" localid="1649358674905" $$\begin{gathered} \frac{d{y}_1}{dx}=k{y}_1 \\ \frac{d{y}_2}{dx}=k{y}_2 \end{gathered}$$

Add the two equations obtained above

$\frac{d{y}_1}{dx}+\frac{d{y}_2}{dx}=k{y}_1+k{y}_2$

The above relation is the condition that the function y, (x)+ y, (x) satisfies the differential

equation $\frac{dy}{dx}=ky$Hence, If two functions are solutions of a differential equation, then their sum dx will also be a solution of the same differential equation.