y%3Dxe%5Ex-sin-x.jpeg "(d^2+3d+2)y=xe^x Sin X")
Solving the Differential Equation (d^2 + 3d + 2)y = xe^x sin x
In this article, we will discuss how to solve the differential equation (d^2 + 3d + 2)y = xe^x sin x. This equation is a second-order linear differential equation with variable coefficients.
Step 1: Divide both sides by the coefficient of the highest derivative term
To start, we will divide both sides of the equation by the coefficient of the highest derivative term, which is 1. This gives us:
(d^2 + 3d + 2)y = xe^x sin x
Step 2: Use the annihilator method to eliminate the exponential term
Notice that the right-hand side of the equation contains an exponential term e^x. To eliminate this term, we can use the annihilator method.
The annihilator of e^x is the operator (D - 1), where D is the derivative operator. Applying this operator to both sides of the equation, we get:
(D - 1)(d^2 + 3d + 2)y = (D - 1)xe^x sin x
Step 3: Simplify the equation
Simplifying the equation, we get:
d^2y + 2dy + y = x sin x
Step 4: Use the method of undetermined coefficients to find the particular solution
To find the particular solution, we will use the method of undetermined coefficients.
Assume that the particular solution has the form:
y_p = A x^2 sin x + B x^2 cos x + C x sin x + D x cos x
Substituting this into the equation, we get:
d^2y_p + 2dy_p + y_p = x sin x
Comparing the coefficients of x sin x and x^2 sin x, we get:
A = -1/4, B = 0, C = 1/4, D = 0
So, the particular solution is:
y_p = -1/4 x^2 sin x + 1/4 x sin x
Step 5: Find the general solution
To find the general solution, we need to find the homogeneous solution y_h and add it to the particular solution y_p.
The homogeneous equation is:
d^2y + 2dy + y = 0
Solving this equation, we get:
y_h = c1 e^(-x) + c2 e^(-2x)
So, the general solution is:
y = y_p + y_h = -1/4 x^2 sin x + 1/4 x sin x + c1 e^(-x) + c2 e^(-2x)
Conclusion
In this article, we have shown how to solve the differential equation (d^2 + 3d + 2)y = xe^x sin x using the annihilator method and the method of undetermined coefficients. The general solution is a combination of the particular solution and the homogeneous solution.
