Final answer:
The general solution to the nonhomogeneous differential equation y''+3y'+2y = 4x² is Y = C₁e⁻ˣ+C₂e⁻²ˣ+7-6x+2x². To find the general solution, we first solve the corresponding homogeneous equation y''+3y'+2y = 0 by finding the roots of the characteristic equation. Next, we find a particular solution yp to the nonhomogeneous equation by assuming a particular solution of the form Ax²+Bx+C and substituting it into the differential equation.
Step-by-step explanation:
The general solution to the nonhomogeneous differential equation y''+3y'+2y = 4x² is:
Y = C₁e⁻ˣ+C₂e⁻²ˣ+7-6x+2x²
To find the general solution, we first solve the corresponding homogeneous equation y''+3y'+2y = 0 by finding the roots of the characteristic equation:
r²+3r+2 = 0
The roots of the characteristic equation are -1 and -2, so the general solution to the homogeneous equation is yh = C₁e⁻ˣ+C₂e⁻²ˣ.
Next, we find a particular solution yp to the nonhomogeneous equation. Since the nonhomogeneous term is 4x², a polynomial of degree 2, we assume a particular solution of the form yp = Ax²+Bx+C. Substituting this into the differential equation, we get:
2A + 6Ax + 4B + 3A + 6B + 2C = 4x²
Equating the coefficients of like powers of x, we get:
A = 1, B = -3, C = 7
So a particular solution to the nonhomogeneous equation is yp = x²-3x+7.
The general solution to the nonhomogeneous differential equation is then given by:
Y = yh + yp = C₁e⁻ˣ+C₂e⁻²ˣ+x²-3x+7.