Question

In Problems 1–22 Solve the given differential equation by separation of variables.

$\frac{\mathbf{\text{dy}}}{\mathbf{\text{dx}}}\mathbf{\text{=}}{\left(\frac{{\displaystyle \text{2y+3}}}{{\displaystyle \text{4x+5}}}\right)}^{\mathbf{\text{2}}}$

Short answer

$y=\frac{(4x+5)}{1+C_1(4x+5)}-\frac{3}{2}$

Step-by-step solution

Step 1:Definition of separable equation

A first-order differential equation of the form $\frac{dy}{dx}=g\left(x\right)h\left(y\right)$is said to be separable or to have separable variables

Separate variables and integration

Separate the variables

$\begin{array}{c}\int \frac{dy}{{(2y+3)}^2}=\int \frac{dx}{{(4x+5)}^2}\\ \frac{-1}{2(2y+3)}=\frac{-1}{4(4x+5)}+C\\ \frac{2}{(2y+3)}=\frac{1}{(4x+5)}+C_1\end{array}$

Where

$C_1=-4C$

Hence,

$\begin{array}{c}2y+3=\frac{2(4x+5)}{1+C_1(4x+5)}\\ y=\frac{(4x+5)}{1+C_1(4x+5)}-\frac{3}{2}\end{array}$

Therefore the solution is$\text{y=}\frac{\text{(4x+5)}}{{\text{1+C}}_{\text{1}}\text{(4x+5)}}\text{-}\frac{\text{3}}{\text{2}}$