Question

y'' + y' - 6y = 2x

Answer 1

Solution.

$$y''+y'-6y=2x.$$

This is a linear differential equation of the second-order with constant coefficients. The general solution of this equation is found as the sum of the general solution $\tilde{y}$ ​ of the corresponding homogeneous equation and some particular integral solution $y_*​$ of the inhomogeneous equation:$y=\tilde{y}+y_*.$

Сompose and solve the characteristic equation:

$$\lambda^2+\lambda-6=0, \newline \lambda_1=-3, \lambda_2=2.$$

 Therefore, $\tilde{y}=C_1e^{\lambda_1x}+C_2e^{\lambda_2x}.$ So,

$$\tilde{y}=C_1e^{-3x}+C_2e^{2x}.$$

Since the right-hand side also contains a $2x$ , we find a particular integral solution $y_*$ ​ ​in the form: 

$$y_*=Ax+B.$$

Using the method of undetermined coefficients we find that $A=-\frac{1}{3}, B=-\frac{1}{18}.$

So, the particular integral solution of equation $y''+y'-6y=2x$ is

$$y_*=-\frac{1}{3}x-\frac{1}{18}.$$

And the general solution of the same equation is

$$y=C_1e^{-3x}+C_2e^{2x}-\frac{1}{3}x-\frac{1}{18}.$$

Answer. $y=C_1e^{-3x}+C_2e^{2x}-\frac{1}{3}x-\frac{1}{18}.$