Solve the differential equation dy/dx = ((2y + 3)/(4x + 5))^2.

Step-by-step solution

1. 1

   Recognise the separable structure. Squaring the quotient keeps the variables apart:

   $\frac{dy}{dx}=\frac{(2y+3)^{2}}{(4x+5)^{2}},$

   so all the $y$ material can go on one side and all the $x$ material on the other.

2. 2

   Separate. Divide by $(2y+3)^2$ and multiply by $dx$:

   $\frac{dy}{(2y+3)^{2}}=\frac{dx}{(4x+5)^{2}}.$

   Dividing by $(2y+3)^2$ assumes it is non-zero — set that case aside for the last step.

3. 3

   Integrate both sides. Each is a power rule with a linear inside function, so divide by the inside coefficient:

   $\int (2y+3)^{-2}dy=-\frac{1}{2(2y+3)},\qquad \int (4x+5)^{-2}dx=-\frac{1}{4(4x+5)}.$

4. 4

   Equate and tidy the constant.

   $-\frac{1}{2(2y+3)}=-\frac{1}{4(4x+5)}+C_0.$

   Multiplying through by $-4$ and renaming the constant gives the cleaner implicit form

   $\frac{2}{2y+3}=\frac{1}{4x+5}+C.$

5. 5

   Check the result by differentiating a member of the family. Taking $C=\tfrac13$ and solving for $y$ gives $y=\dfrac{3(2x+1)}{4(x+2)}$; differentiating this and comparing with $\left(\dfrac{2y+3}{4x+5}\right)^2$ leaves a difference that simplifies to exactly $0$, so the implicit solution is correct.

6. 6

   Restore the solution lost during separation. The constant function $y=-\tfrac32$ makes $2y+3=0$, so $\dfrac{dy}{dx}=0$ on both sides — it is a genuine (singular) solution, but no finite $C$ produces it. Always check the values you divided by.