Question

In Problems 7–12 match each of the given differential equations with one or more of these solutions:

(a) $\mathit{y}\mathbf{=}\mathbf{0}$, (b) $\mathit{y}\mathbf{=}\mathbf{2}$, (c)$\mathit{y}\mathbf{=}\mathbf{2}\mathit{x}$, (d) $\mathit{y}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{2}}$

$\mathit{x}\mathit{y}\mathbf{"}\mathbf{-}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{0}$

Short answer

Answer:

The solutions are $y=0,y=2$, and $y=2x^2$.

Step-by-step solution

Check y=0 if is a solution.

Put $y=0$.

$$\begin{gathered} x\left(0\right)"-\left(0\right)\text{'}=0 \\ 0-0=0 \\ 0=0 \end{gathered}$$

Hence, it is a solution.

Check y=2 if is a solution.

Put $y=2$.

$$\begin{gathered} x\left(2\right)"-\left(2\right)\text{'}=0 \\ 0-0=0 \\ 0=0 \end{gathered}$$

Hence, it is a solution.

Check y=2x if is a solution.

Put $y=2x$.

$$\begin{gathered} x\left(2x\right)"-\left(2x\right)\text{'}=0 \\ x\left(2\right)\text{'}-2=0 \\ -2\ne 0 \end{gathered}$$

Hence, it is not a solution.

Check y=2x2 if is a solution.

Put $y=2x^2$.

role="math" localid="1655701754393" $$\begin{gathered} x\left(2x^2\right)"-\left(2x^2\right)\text{'}=0 \\ x\left(4x\right)-4x=0 \\ x\left(4\right)-4x=0 \\ 0=0 \end{gathered}$$

Hence, it is a solution.