Definition 1 document [1]:? A functional equation containing independent variables, unknown functions and derivatives (or differentials) of unknown functions is called a differential equation, and the highest order number of derivatives of unknown functions appearing in the differential equation is called the order of the differential equation.

So this is a second-order differential equation, which we generally call a second-order non-homogeneous linear equation with constant coefficients.

Definition 2 [1]: If the solution of a differential equation contains n independent arbitrary constants, it is called the general solution of the differential equation. If any constant in the general solution is determined, the special solution of the differential equation can be obtained.

This problem requires that the special solution must not contain any constants.

Ideas:

1. Find the general solution first; 2. Seek a special solution again.

Method:

Because of the structure of solutions of second-order non-homogeneous linear equations with constant coefficients: reference [3]

General solution of second-order homogeneous linear equation with constant coefficients = general solution of second-order homogeneous linear equation with constant coefficients+1 special solution of second-order homogeneous linear equation with constant coefficients.

therefore

Firstly, the general solution of the second-order homogeneous linear equation with constant coefficients is found [2]

2.？ Then find 1 special solutions of the second-order non-homogeneous linear equation with constant coefficients. Literature [4]

Means of finding homogeneous solution: write characteristic equation, find characteristic root, and then determine formula.

The method of finding non-homogeneous special solution: undetermined coefficient method

Solution: 1 Find the homogeneous general solution y.

Write the characteristic equation: r? + 1=0

r 1=i，r2=-i

therefore

General solution Y=c 1cosx+c2sinx.

2. Homogeneous special solution y*

Let the special solution form be y * = AE (-x)

y*'=-ae^(-x),y*''=ae^(-x)

Substituting into the original equation, you get

ae^(-x)+ae^(-x)=3e^(-x)

2a=3

a=3/2

therefore

y*=3/2e^(-x)

therefore

Non-homogeneous general solution y = y+y * = c1cosx+c2sinx+3/2e (-x)

3. In a known way

Y(0)=0，y (0)=0，get

c 1+3/2=0①

y'=-c 1sinx+c2cosx-3/2e^(-x)

0=c2-3/2②

From (1), we get

c 1=-3/2

From 2, get

c2=3/2

therefore

The special solution y =-3/2cosx+3/2sinx+3/2e (-x).

References:

Advanced Mathematics, Department of Mathematics, Tongji University, Higher Education Press, 1-6.

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