y%3De%5E2x%2Bx%5E2%2Bsin3x.jpeg "(d^2-4d+4)y=e^2x+x^2+sin3x")
Solving the Differential Equation (d^2-4d+4)y=e^2x+x^2+sin3x
In this article, we will solve the differential equation (d^2-4d+4)y=e^2x+x^2+sin3x. This is a second-order linear nonhomogeneous differential equation with constant coefficients.
Step 1: Find the Complementary Solution
The first step in solving a nonhomogeneous differential equation is to find the complementary solution, which is the solution to the associated homogeneous equation. The homogeneous equation is:
(d^2-4d+4)y = 0
To find the complementary solution, we need to find the roots of the characteristic equation:
r^2 - 4r + 4 = 0
Factoring the left-hand side, we get:
(r - 2)^2 = 0
This gives us a repeated root of r = 2. Therefore, the complementary solution is:
y_c = c1e^(2x) + c2xe^(2x)
where c1 and c2 are constants.
Step 2: Find a Particular Solution
To find a particular solution, we need to find a function y_p that satisfies the nonhomogeneous equation:
(d^2-4d+4)y_p = e^2x + x^2 + sin3x
We can use the method of undetermined coefficients to find a particular solution. Let's assume that the particular solution has the form:
y_p = Ae^2x + Bx^2 + Cx + Dsin3x + Ecos3x
where A, B, C, D, and E are constants.
Substituting this into the nonhomogeneous equation, we get:
(4A - 8B) e^2x + (2B - 4C) x^2 + (-4B + 2C - 9D) x + (2C - 9E) sin3x + (-9D) cos3x = e^2x + x^2 + sin3x
Equating the coefficients of like terms, we get:
4A - 8B = 1 2B - 4C = 1 -4B + 2C - 9D = 0 2C - 9E = 1 -9D = 0
Solving this system of equations, we get:
A = 1/4 B = -1/8 C = 1/16 D = 0 E = 1/144
Therefore, the particular solution is:
y_p = (1/4)e^2x - (1/8)x^2 + (1/16)x + (1/144)cos3x
Step 3: Find the General Solution
The general solution is the sum of the complementary solution and the particular solution:
y = y_c + y_p = c1e^(2x) + c2xe^(2x) + (1/4)e^2x - (1/8)x^2 + (1/16)x + (1/144)cos3x
This is the general solution to the differential equation (d^2-4d+4)y=e^2x+x^2+sin3x.
