y%3Dxsinx%2Bx%5E2e%5Ex.jpeg "(d^2-1)y=xsinx+x^2e^x")
Solving the Differential Equation (d^2-1)y=xsinx+x^2e^x
In this article, we will explore the solution to the differential equation (d^2-1)y=xsinx+x^2e^x. This equation is a second-order linear nonhomogeneous differential equation with variable coefficients.
Step 1: rewrite the equation
First, let's rewrite the equation in a more familiar form:
y'' - y = xsinx + x^2e^x
Step 2: find the homogeneous solution
The homogeneous solution is obtained by setting the right-hand side equal to zero and solving the resulting homogeneous equation:
y'' - y = 0
The general solution to this equation is:
y_h = c1e^x + c2e^-x
where c1 and c2 are arbitrary constants.
Step 3: find a particular solution
To find a particular solution, we need to find a function y_p that satisfies the original equation:
y_p'' - y_p = xsinx + x^2e^x
Using the method of undetermined coefficients, we assume a particular solution of the form:
y_p = Axsinx + Bx^2e^x + Cxsinx + Dx^2e^x
Substituting this into the equation and equating coefficients, we get:
A = -1, B = 1, C = 0, D = 1/2
So, the particular solution is:
y_p = -xsinx + x^2e^x + (1/2)x^2e^x
Step 4: find the general solution
The general solution is the sum of the homogeneous solution and the particular solution:
y = y_h + y_p = c1e^x + c2e^-x - xsinx + x^2e^x + (1/2)x^2e^x
Conclusion
In this article, we have successfully solved the differential equation (d^2-1)y=xsinx+x^2e^x using the method of undetermined coefficients. The general solution is a linear combination of the homogeneous solution and the particular solution.
Note: The constants c1 and c2 can be determined by applying the initial conditions or boundary conditions of the problem.
