Question

If${\mathit{y}}_{\mathbf{1}}\mathbf{=}{\mathit{e}}^{\mathbf{x}}$and${\mathit{y}}_{\mathbf{2}}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{x}}$are solutions of homogeneous lineardifferential equation, then necessarily$\mathit{y}\mathbf{=}\mathbf{-}\mathbf{5}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathbf{+}\mathbf{10}{\mathit{e}}^{\mathbf{x}}$is alsoa solution of the DE. _______

Short answer

Yes, the solution of an DE is${\text{y=-5e}}^{\text{-x}}{\text{+10e}}^{\text{x}}$

Step-by-step solution

Step 1:Definea solution of the differential equation.

A differential equation solution is a statement for the dependent variable in terms of one or more independent variables that fulfils the relation. All feasible solutions are included in the general solution, which mainly contains arbitrary constants or arbitrary functions.

A solution of the differential equation is y=-5e-x+10ex.

Yes.

Let the solutions of a homogeneous linear differential equation be,

$$\begin{gathered} {\text{y}}_{\text{1}}{\text{=e}}^{\text{x}} \\ {\text{y}}_{\text{2}}{\text{=e}}^{\text{-x}} \end{gathered}$$

Then, a solution for this differential equation is,

${\text{y=-5e}}^{\text{-x}}{\text{+10e}}^{\text{x}}$

Which in the form.${\text{y=c}}_{\text{1}}{\text{y}}_{\text{1}}{\text{+c}}_{\text{2}}{\text{y}}_{\text{2}}$

So, the value of constants are ${\text{c}}_{\text{1}}\text{=10}$and .${\text{c}}_{\text{2}}\text{=-5}$