How to solve this nonlinear differential equation

[Brucezhou]

dy/dx=2x+y^2
By the way, methods of solving linear differential equation are useless, such as integrating factor and Bernoulli method.

 

[Mark44]

The OP is a high school student who has worked a bit with linear differential equations. The nonlinear equation above is one that he cooked up for himself.

 

[LCKurtz]

dy/dx=2x+y^2
By the way, methods of solving linear differential equation are useless, such as integrating factor and Bernoulli method.

Do you have a question, or are you just pontificating?

 

[Brucezhou]

Do you have a question, or are you just pontificating?

The question is to find the function involving variables x and y.

 

[Ray Vickson]

The question is to find the function involving variables x and y.

The solution involves "non-elementary" functions. Solving it in Maple results in

y(x) = 2^(1/3)*(c*AiryAi(1, -2^(1/3)*x)
+AiryBi(1, -2^(1/3)*x))/(c*AiryAi(-2^(1/3)*x)+AiryBi(-2^(1/3)*x)),

where AiryA and AiryB are the so-called "Airy Wave Functions" and c is a constant. AiryA(z) and AiryB(z) are two linearly independent solutions of the differential equation
\frac{d^2 w}{dz^2} - z w = 0.
They can be written in terms of hypergeometric functions.

 

[Brucezhou]

I think I have to explain what I said in the question. Bernoulli method and integrating factor don't work since I've tried several times but failed. And now I don't want others to waste too much time in these two ways, so I typed those words. But if someone finds this question can be solved using these two methods, I will be really happy since those are only two of a few ways I learned for differential equations. What I did using Bernoulli is

dy/dx=2x+y^2

Then divided by y^2 on both sides

y^2*dy/dx=2x/y^2+1

y^2*dy/dx-2x/y^2=1

Then use du=1/y^2*dy replacing all the elements about y in the equation

u=-1/y

du/dx-2xu^2=1

Then I don't know what I can do.

 

[JJacquelin]

I think I have to explain what I said in the question. Bernoulli method and integrating factor don't work since I've tried several times but failed. And now I don't want others to waste too much time in these two ways, so I typed those words. But if someone finds this question can be solved using these two methods, I will be really happy since those are only two of a few ways I learned for differential equations.


What I did using Bernoulli is

dy/dx=2x+y^2

Then divided by y^2 on both sides

y^2*dy/dx=2x/y^2+1

y^2*dy/dx-2x/y^2=1

Then use du=1/y^2*dy replacing all the elements about y in the equation

u=-1/y

du/dx-2xu^2=1

Then I don't know what I can do.

What you do is not the usual way to solve a Bernoulli ODE.
This one is classical. Just let y(x)=-(df/dx)/f(x)
This leads to the second order linear ODE : f ''+2xf(x) = 0 which is an Airy ODE (related to Bessel ODEs)

 

[LCKurtz]

I think I have to explain what I said in the question. Bernoulli method and integrating factor don't work since I've tried several times but failed. And now I don't want others to waste too much time in these two ways, so I typed those words. But if someone finds this question can be solved using these two methods, I will be really happy since those are only two of a few ways I learned for differential equations.

Most differential equations are not solvable exactly. Sometimes, for those that are, methods have been discovered that will solve a particular class of DE's. Introductory textbooks teach some of those types of DE's and the methods that solve them. This sometimes gives students the false impression that these methods are more useful than they are or that somehow, all DE's can be solved if you just know the right trick.

You can't expect methods that may work on some first order DE's or some linear DE's to work on higher order or nonlinear DE's. The DE in your example is solvable by methods as suggested by others, but you were just lucky there. Write down a slightly more complicated one and you will likely not find any method to solve it exactly.