Solve the differential equation dy/dx = ((2y + 3)/(4x + 5))^2 and give the general solution in implicit form.

Step-by-step solution

1. 1

   Check that the equation is separable before choosing a method. Squaring the quotient keeps the $y$-part and the $x$-part in separate factors:

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

   which is exactly the form $y'=g(y)h(x)$. No integrating factor is needed — this is not linear in $y$, so trying the linear-ODE machinery here would fail.

2. 2

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

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

   Dividing by $(2y+3)^{2}$ costs you the constant solution $y=-\tfrac32$, which does satisfy the original equation ($y'=0$ and the right side is $0$). It is a singular solution outside the general family, so note it now.

3. 3

   Integrate the $y$-side with the substitution $u=2y+3$. Then $du=2\,dy$, so $dy=\tfrac{du}{2}$ and

   $\int\frac{dy}{(2y+3)^{2}}=\frac12\int u^{-2}\,du=-\frac{1}{2u}=-\frac{1}{2(2y+3)}.$

   The factor $\tfrac12$ comes from the chain rule; forgetting it is the single most common error in linear-substitution integrals.

4. 4

   Integrate the $x$-side the same way with $v=4x+5$. Here $dv=4\,dx$, so

   $\int\frac{dx}{(4x+5)^{2}}=\frac14\int v^{-2}\,dv=-\frac{1}{4v}=-\frac{1}{4(4x+5)}.$

   The constant out front is $\tfrac14$ because the inner derivative is $4$, matching the pattern $\int f(ax+b)\,dx=\tfrac1a F(ax+b)$.

5. 5

   Combine and absorb the constants. Equating the two antiderivatives and adding one arbitrary constant:

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

   Multiplying by $-2$ and renaming the constant gives the tidier implicit form

   $\frac{1}{2y+3}=\frac{1}{8x+10}+C,$

   since $2\cdot 4(4x+5)=8x+10$ after the factor of $2$ is distributed.

6. 6

   Verify by implicit differentiation. Differentiating $\frac{1}{2y+3}=\frac{1}{8x+10}+C$ with respect to $x$:

   $-\frac{2y'}{(2y+3)^{2}}=-\frac{8}{(8x+10)^{2}}=-\frac{8}{4(4x+5)^{2}},$

   so $y'=\frac{(2y+3)^{2}}{(4x+5)^{2}}$ ✓. A numerical spot check with $C=0.3$ at $x=1$ gives $y'=0.09765625$ from the implicit solution and $0.09765625$ from the right-hand side.