Question

Verify that the piecewise-defined function $\left\{\begin{array}{l}-x^2,x<0\\ x^2,x⩾0\end{array}\right.$is a solution of the differential equation$xy\text{'}-2y=0$ on$(-\infty ,\infty )$.

Short answer

The piecewise-defined function is a solution of the differential equation.

Step-by-step solution

Determine the limits of the function y=-x2.

Check the continuity of the function at$x<0$.

The first derivative of the function is,

$y\text{'}=-2x$

Substitute $y$and$y\text{'}$ in the differential equation.

$$\begin{gathered} x(-2x)-2(-x^2)=0 \\ 0=0 \end{gathered}$$

Hence, the left-hand limit is equal to the right-hand limit, so the function is continuous at $x<0$and the solution is verified.

Determine the limits of the function y=x2.

Check the continuity of the function at$x⩾0$.

The first derivative of the function is,

$y\text{'}=2x$

Substitute and in the differential equation.

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

Hence, the left-hand limit is equal to the right-hand limit, so the function is continuous at $x⩾0$and the solution is verified.

Hence, the piecewise-defined function is a solution of the differential equation on the interval $(-\infty ,\infty )$.