Chapter 6, Problem 1RE

Determining a Solution Determine whether the function $y=x^3$ is a solution of the differential equation $2x{y}^{\prime }+4y=10x^3$.

To determine

To calculate: Whether $y=x^3$ is a solution of the given differential equation $2xy\text{'}+4y=10x^3$.

Answer to Problem 1RE

Solution:

$y=x^3$ is a solution of $2xy\text{'}+4y=10x^3$

Explanation of Solution

Given:

Function $y=x^3$

Differential equation $2xy\text{'}+4y=10x^3$

Formula used:

$\frac{d}{dx}(x^n)=n{x}^{n-1}$

Calculation:

Differentiate the function $y=x^3$ with respect to $x$ to get

$y\text{'}=\frac{dy}{dx}=3x^2$

Put the value of $y$ and $y\text{'}$ in left hand side of differential equation

$\begin{array}{ccc}\hfill 2xy\text{'}+4y& =& 2x\left(3x^2\right)+4x^3\hfill \\ \hfill & =& 6x^3+4x^3\hfill \\ \hfill & =& 10x^3\hfill \end{array}$

The above result shows that the left side of the equality is equal to the right side.

Thus, it can be concluded that $y=x^3$ is solution of differential equation $2xy\text{'}+4y=10x^3$.