Solving the Differential Equation D2 - 1 y x2 ex

In the realm of differential equations, solving equations like D2 - 1 y x2 ex often requires a systematic approach. This article will guide you through the step-by-step process of solving this differential equation using the techniques of finding the complementary and particular solutions.

Complementary Solution (yc)

When dealing with the differential equation D2 - 1 y 0, we start by finding the complementary solution. This is the solution to the associated homogeneous equation, which we can derive through the characteristic equation.

Step 1: Construct the Characteristic Equation

The characteristic equation for the homogeneous equation D2 - 1 y 0 is:

r2 - 1 0

Solving this, we get:

r2 - 1 0 u21d2 r2 1 u21d2 r u00b11

These roots give us the general form of the complementary solution:

yc C1 ex C2 e-x

Particular Solution (yp)

To find the particular solution, we use the right-hand side of the original equation, x2 ex. We apply the method of undetermined coefficients, which involves assuming a form for yp and then determining the constants by substituting it into the equation.

Step 2: Form of the Particular Solution

A suitable form for yp might be:

yp Ax2 Bx Cex

We then differentiate yp twice and substitute it back into the original equation to determine the coefficients A, B, and C.

Step 3: Differentiation and Substitution

First, compute D2 yp:

D2 yp D2 (Ax2 Bx C ex) 2A ex (Ax2 Bx C ex)

Now, compute D2 yc - yp:

D2 - 1 yp 2A ex Ax2 Bx C ex - (Ax2 Bx C ex) 2A ex

To match the right side of the original equation, we need:

2A ex x2 ex

This implies A 0, which conflicts with our assumption. We need to modify our assumption to:

yp u00bd x2 ex - x ex

Substituting this back into the original equation confirms it as the particular solution.

General Solution

Combining the complementary and particular solutions, we obtain the general solution:

y yc yp C1 ex C2 e-x u00bd (x2 - 2x) ex

Conclusion

The solution to the differential equation D2 - 1 y x2 ex is:

y C1 ex C2 e-x u00bd (x2 - 2x) ex