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Freshman mathematical differential equation
Solution: The following are the basic steps to solve the second-order linear non-homogeneous differential equation with constant coefficients:

1 Find the general solution of the corresponding homogeneous equation first;

The first step is to write the characteristic equation of homogeneous differential equation: r 2+4r+3 = 0.

Step 2, find two roots of the characteristic equation: r 1=- 1, r2=-3.

The third step is to determine the general solution of homogeneous differential equation: because the characteristic equation has two unequal real roots, the general solution is y = c 1 * e (-x)+c2 * e (-3x).

2. Find the special solution of nonhomogeneous equation;

Since the nonhomogeneous term f (x) = e x belongs to the type of f (x) = pm (x) * e (λ x), and m=0, λ= 1 (non-characteristic root).

Therefore, the special solution of non-homogeneous equation can be set as: y0 = x 0 * a * e x = a * e x

Substituting into the nonlinear equation, we get: (1+4+3) * a * e x = e x, and the solution is: A= 1/8.

So the special solution is: y0 =1/8 * e x.

To sum up, the general solution of the second-order linear non-homogeneous differential equation with constant coefficients is:

Y = c 1 * e (-x)+C2 * e (-3x)+1/8 * e x (where c1and C2 are arbitrary constants).