Question

Solve dy/dx=(4x+y+1)^2
​

Answer 1

dydx=(4x+y+1)2

Use a substitution to replace the quantity squared.

Let u=(4x+y+1) and dudx=4+dydx

dudx−4=u2

dudx=(u2+4)

This is a separable ODE. Form the two integrals and solve them.

∫du(u2+4)=∫dx

x=∫du(u2+4)

The remaining integral requires a trigonometric substitution.

x=∫du(u2+4)

u=2tan(θ),du=2sec2(θ)dθ

x=∫2sec2(θ)4(tan2(θ)+1)dθ

x=∫2sec2(θ)4sec2(θ)dθ

x=∫12dθ=12θ+C

Change variables from θ back to u.

x=12tan−1(u2)+C

Then, change variables from u back to x,y.

x=12tan−1((4x+y+1)2)+C

Solve the equation for y to get the answer.

Answer

y=2tan(2x+B)−4x−1