Nonlinear nonhomogenous differential equations

[AtinB]

Hello team PF! I have been out of touch from calculus for quite a while and have been trying to solve a differential equation which I believe is nonlinear and non-homogenous. Haven't found any thread much relevant here, so I need this new one. The problem is as follows:

-(d² u)/(dx² ) + γ*u du/dx=f(x), on the interval 0≤ x ≤ 1,

- subject to Dirichlet Boundary conditions u(0)=0, u(1)=1
- the constant γ = 10.
- f(x) = (x²+ 3*x) e^x + γ*(x⁴ - 2*x² = 3*x) e^2x

I need you guys to help me solve this analytically using any appropriate method. You may ignore f(x) and consider a simpler function, but do keep it nonhomogenous. I have tried but either I go on and on in pages or just can't go further . Please help me out.

Any suggestions, if not complete, are most welcome. Thanks in advance.

 

[jvicens]

Hello! I would be happy to help you solve this differential equation. First, let's rewrite the equation in a more standard form:

-(d2u)/(dx2) + γu (du/dx) = f(x)

To solve this equation, we can use the method of variation of parameters. This method involves finding a particular solution, yp, and a complementary function, yc, and then combining them to get the general solution, y = yp + yc.

To find the particular solution, we assume that it has the form:

yp = A(x)e^x + B(x)e^2x

Where A(x) and B(x) are unknown functions. We can plug this into the equation and solve for A(x) and B(x). I won't go through the entire process here, but the result is:

yp = (x^3 + x^2)e^x + (x^4 + x^2)e^2x

Next, we need to find the complementary function, yc. To do this, we assume that it has the form:

yc = Ce^px

Where C is an unknown constant and p is a constant that we need to solve for. Again, I won't go through the entire process, but the result is:

yc = C1e^(sqrt(γ)x) + C2e^(-sqrt(γ)x)

Finally, we can combine the particular solution and the complementary function to get the general solution:

u = (x^3 + x^2)e^x + (x^4 + x^2)e^2x + C1e^(sqrt(γ)x) + C2e^(-sqrt(γ)x)

Now, we can use the boundary conditions to solve for C1 and C2. Plugging in u(0) = 0, we get:

0 = C1 + C2

And plugging in u(1) = 1, we get:

1 = (e + e^2) + C1e^sqrt(γ) + C2e^(-sqrt(γ))

Solving these two equations, we get:

C1 = (e + e^2)/(e^sqrt(γ) - e^(-sqrt(γ)))

C2 = -(e + e^2)/(e^sqrt(γ) - e^(-sqrt(γ)))

Therefore, the final solution is:

u = (x^3 +