Find the general solution of the following ODE

Question

Find the general solution of the following ODE:

      .

Compute all coefficients of the particular solution.

Answer 1

Because Characteristic equation of homogenius equation: r² - 1 = 0, r = ± 1.

Then general solution of homogenius solution is y_h = C₁e^x :+ C₂e^-x.

Particular solution of given equation is y_p = (ax + b)e^2x, where a and b is unknown.

Now y_p' = ae^2x + 2(ax + b)e^2x = (2ax + 2b + a)e^2x; y_p^'' = 2ae^2x + 2(2ax + 2b + a)e^2x = (4ax + 4b + 4a)e^2x.

Substitution in given equation after reducing by e^2x we get: 4ax + 4b + 4a - ax - b = x or 3ax + 3b + 4a = x.

So, a = 1/3, 3b + 4/3 = 0, b = - 4/9and because y_p = (1/3x - 4/9)e^2x and general solution of given ODE is:

y = C₁e^x + C₂e^-x + (1/3x - 4/9)e^2x