y%3Dx%5E3.jpeg "(d^2-4d+4)y=x^3")
Solving the Differential Equation (d^2-4d+4)y=x^3
In this article, we will solve the differential equation (d^2-4d+4)y=x^3. This is a second-order linear differential equation with a polynomial term on the right-hand side.
Step 1: Homogeneous Solution
To solve this differential equation, we need to find the homogeneous solution first. The homogeneous equation is obtained by setting the right-hand side to zero:
(d^2-4d+4)y=0
This is a homogeneous linear differential equation with constant coefficients. The characteristic equation is:
r^2 - 4r + 4 = 0
Solving for r, we get:
r = 2
Since the roots are identical, the homogeneous solution is:
y_h = (c1 + c2x)e^(2x)
where c1 and c2 are arbitrary constants.
Step 2: Particular Solution
To find the particular solution, we need to find a function that satisfies the original differential equation. Let's assume a particular solution of the form:
y_p = Ax^3 + Bx^2 + Cx + D
Substituting this into the original differential equation, we get:
(d^2-4d+4)(Ax^3 + Bx^2 + Cx + D) = x^3
Expanding and collecting like terms, we get a system of equations:
A = 1/4
B = 0
C = 0
D = 0
Thus, the particular solution is:
y_p = (1/4)x^3
Step 3: General Solution
The general solution is the sum of the homogeneous and particular solutions:
y = y_h + y_p
y = (c1 + c2x)e^(2x) + (1/4)x^3
This is the general solution to the differential equation (d^2-4d+4)y=x^3.
Conclusion
In this article, we have solved the differential equation (d^2-4d+4)y=x^3 using the method of homogeneous and particular solutions. The general solution is a combination of the exponential and polynomial terms.
