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) $\mathit{y}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{2}}$

$\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathit{y}$

Short answer

Answer:

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

Step-by-step solution

Check y=0if is a solution.

Put $y=0$.

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

Hence, it is a solution.

Check y=2 if is a solution.

Put $y=2$.

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

Hence, it is not a solution.

Check y=2x if is a solution.

Put $y=2x$.

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

Hence, it is a solution.

Check y=2x2 if is a solution.

Put $y=2x^2$.

$$\begin{gathered} x\left(2x^2\right)\text{'}=2x^2 \\ x\left(4x\right)=2x^2 \\ 4x^2\ne 2x^2 \end{gathered}$$

Hence, it is not a solution.