y%3Dx%5E2sinx.jpeg "(d^2-4d+4)y=x^2sinx")
Solving the Differential Equation (d^2-4d+4)y=x^2sinx
In this article, we will discuss the solution to the differential equation (d^2-4d+4)y=x^2sinx. This equation is a second-order linear differential equation with a non-constant coefficient.
Method of Undetermined Coefficients
To solve this equation, we can use the method of undetermined coefficients. This method involves assuming a particular solution of the form y_p = x^2(A sinx + B cosx), where A and B are constants to be determined.
Step 1: Find the Homogeneous Solution
The homogeneous equation is (d^2-4d+4)y=0, which has a general solution of the form y_c = c1 e^(2x) + c2 e^(2x). However, since the given equation is a non-homogeneous equation, we need to find a particular solution that satisfies the given equation.
Step 2: Find the Particular Solution
Assume a particular solution of the form y_p = x^2(A sinx + B cosx). Substituting this into the given equation, we get:
(d^2-4d+4)(x^2(A sinx + B cosx)) = x^2sinx
Expanding the left-hand side, we get:
(-4A + 2B)x sinx + (4A + 2B)x cosx - 4A x^2 sinx - 4B x^2 cosx = x^2sinx
Comparing coefficients, we get:
-4A + 2B = 1 ... (1)
4A + 2B = 0 ... (2)
-4A = 0 ... (3)
-4B = 0 ... (4)
Solving these equations, we get A = 1/4 and B = 0. Therefore, the particular solution is y_p = (1/4)x^2sinx.
Step 3: Find the General Solution
The general solution is the sum of the homogeneous solution and the particular solution:
y = y_c + y_p
y = c1 e^(2x) + c2 e^(2x) + (1/4)x^2sinx
This is the general solution to the differential equation (d^2-4d+4)y=x^2sinx.
Conclusion
In this article, we have solved the differential equation (d^2-4d+4)y=x^2sinx using the method of undetermined coefficients. The general solution involves the sum of the homogeneous solution and the particular solution.
