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

$\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathit{x}$

Short answer

Answer:

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

Step-by-step solution

Check if  y=0 is a solution

Put $y=0$into the given differential equation and simplify.

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

This solution satisfies the given differential equation. Therefore, it is a solution.

Check if y=2 is a solution

Put $y=2$into the given differential equation and simplify.

$$\begin{gathered} x\left(2\right)\text{'}=2\left(2\right) \\ x\times 0=4 \\ 0\ne 4 \end{gathered}$$

This solution does not satisfy the given differential equation. Therefore, it is not a solution.

Check if y=2x is a solution

Put $y=2x$into the given differential equation and simplify.

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

This solution does not satisfy the given differential equation. Therefore, it is not a solution.

Check if y=2x2 is a solution

Put $y=2x^2$into the given differential equation and simplify.

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

This solution satisfies the given differential equation. Therefore, it is a solution.