How to solve the ODE y' = (4 + y +1)^2

[mohdfasieh]

hello genius guys,
can u people tell me the solution of differential equation dy/dx=(4x+y+1)^2

please replyy
thz in advance

 

[arildno]

Introduce the new function: u(x)=4x+y(x)+1
Then, you have:
$$\frac{du}{dx}=4+\frac{dy}{dx}=4+u^{2}$$
The diff.eq in u is separable.

 

[Architeuthis Dux]

Hello,

To solve this ODE, you can use the method of separation of variables. First, we can rewrite the equation as dy/dx = (4 + y + 1)^2 = (5 + y)^2. Then, we can separate the variables by taking the inverse of both sides and integrating: 1/(5+y)^2 dy = dx.

To integrate the left side, we can use the substitution u = 5 + y, which gives us du = dy. Substituting this into the integral, we get 1/u^2 du = dx. Integrating both sides, we get -1/u = x + C, where C is the constant of integration.

Substituting back in for u, we get -1/(5+y) = x + C. Solving for y, we get y = -5 - 1/(x+C). Therefore, the general solution to the ODE is y = -5 - 1/(x+C), where C is a constant.

I hope this helps! Let me know if you have any further questions.