Question

Using $\left(3.9\right)$, find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $\left(3.9\right)$, and Example 1.

$\mathbf{2}\mathit{x}\mathit{y}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{5}\mathbf{/}\mathbf{2}}$

Short answer

Answer:

The general solution of the differential equation is $y=\frac{x^{5/2}}{3}+\frac{C}{X^{1/2}}$.

Step-by-step solution

Given information

The given differential equation is$2xy+y=2x^{5/2}$.

Meaning of the first-order differential equation

A first-order differential equation is defined by two variables, x and y, and its function $\mathit{f}\left(x,y\right)$is defined on an XY-plane region.

Find the general solution

Write this differential equation to make it in the form $y+Py=Q$; that is

$y+\frac{y}{2x}{x}^{3/2}$.

From equation 3.4,

$$\begin{gathered} I=\int \frac{dx}{2x} \\ =\frac{1}{2}In\left(x\right)e\text{'} \\ =e^{\frac{1}{2}In\left(x\right)} \\ =x^{1/2} \end{gathered}$$.

Find a solution for this differential equation

role="math" localid="1655205234482" $$\begin{gathered} ye\text{'}=\int \left(x^{3/2}\right)x^{1/2}dx \\ =\frac{x^3}{3}+C \\ y=\frac{x^{5/2}}{3}+\frac{C}{x^{1/2}} \end{gathered}$$

Therefore, the general solution of the differential equation is$y=\frac{x^{5/2}}{3}+\frac{C}{x^{1/2}}$.