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Solving the Differential Equation (d^2-1)y=xsinx(1+x^2)e^x
In this article, we will solve the differential equation (d^2-1)y=xsinx(1+x^2)e^x. This equation is a second-order linear ordinary differential equation with variable coefficients. We will use the method of undetermined coefficients to find the general solution of this equation.
Step 1: Write down the homogeneous equation
First, we write down the homogeneous equation corresponding to the given differential equation:
(d^2-1)y = 0
This equation has a general solution of the form:
y_h = c1e^x + c2e^-x
where c1 and c2 are constants.
Step 2: Find a particular solution
Next, we need to find a particular solution of the non-homogeneous equation. We assume that the particular solution has the form:
y_p = x(A sinx + B cosx)(1+x^2)e^x
where A and B are constants to be determined.
Substituting this expression into the original differential equation, we get:
(d^2-1)(x(A sinx + B cosx)(1+x^2)e^x) = xsinx(1+x^2)e^x
Expanding the left-hand side and collecting like terms, we get:
(2A+2B)x^2 sinx + (2A-2B)x cosx + (A-B)sinx + (A+B)cosx = xsinx(1+x^2)e^x
Step 3: Determine the constants A and B
Equating the coefficients of sinx and cosx on both sides, we get:
2A+2B = 1 2A-2B = 0
Solving this system of equations, we get:
A = 1/4 B = 1/4
Step 4: Write down the general solution
The general solution of the differential equation is the sum of the homogeneous solution and the particular solution:
y = c1e^x + c2e^-x + x(1/4 sinx + 1/4 cosx)(1+x^2)e^x
The constants c1 and c2 can be determined by applying the initial conditions.
Conclusion
In this article, we have solved the differential equation (d^2-1)y=xsinx(1+x^2)e^x using the method of undetermined coefficients. The general solution of this equation involves the sum of the homogeneous solution and the particular solution. The constants in the general solution can be determined by applying the initial conditions.
