Answer:
The general solution is the sum of the particular solution and the complementary function: y((x) = (3/4)x + 1/2 + (C1 + C2x)e⁻²ˣ, where C1 and C2 are arbitrary constants.
To solve the given differential equation using the method of undetermined coefficients, assume a particular solution of the form:
y_p(x) = Ax + B
where A and B are constants to be determined.
First, let's find the derivatives of y_p(x):
y'_p(x) = A
y''_p(x) = 0
Now, substitute these derivatives into the original differential equation:
0 + 4(A) + 4(Ax + B) = 3x + 5
Simplifying this equation:
4Ax + 4B + 4A = 3x + 5
Now, equate the coefficients of like terms on both sides of the equation:
4A = 3 (coefficient of x on the right-hand side)
4B + 4A = 5 (constant term on the right-hand side)
Solving these equations simultaneously:
4A = 3
4B + 4A = 5
From the first equation, we find A = 3/4. Substituting this value into the second equation:
4B + 4(3/4) = 5
4B + 3 = 5
4B = 2
B = 1/2
Therefore, the particular solution is:
y_p(x) = (3/4)x + 1/2
To find the general solution, we also need the complementary function. The characteristic equation for the homogeneous equation y'' + 4y' + 4y = 0 is:
r² + 4r + 4 = 0
Factoring this equation, we have:
(r + 2)² = 0
The characteristic equation has a repeated root of -2. Therefore, the complementary function is:
y_c(x) = (C1 + C2x)e⁻²ˣ
where C1 and C2 are constants to be determined.
Hence, the general solution is the sum of the particular solution and the complementary function: y(x) = (3/4)x + 1/2 + (C1 + C2x)e⁻²ˣ , where C1 and C2 are arbitrary constants.