(b) Solve (d^(2)-2d+1)y=x^(2)e^(3x)

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Jul 03, 2024

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(b) Solve (d^(2)-2d+1)y=x^(2)e^(3x)

In this problem, we are asked to solve a second-order linear differential equation with an exponential function on the right-hand side. The differential equation is given as:

(d^(2)-2d+1)y=x^(2)e^(3x)

Step 1: Identify the type of differential equation

The given differential equation is a second-order linear differential equation with constant coefficients. The general form of such an equation is:

a(d^(2)y) + b(dy) + cy = f(x)

where a, b, and c are constants, and f(x) is a function of x.

Step 2: Find the homogeneous solution

The homogeneous solution is the general solution of the associated homogeneous equation:

(d^(2)-2d+1)y=0

Let's find the roots of the characteristic equation:

r^(2) - 2r + 1 = 0

(r - 1)^2 = 0

r = 1 (repeated root)

The homogeneous solution is:

y_h = (c1 + c2x)e^(x)

where c1 and c2 are arbitrary constants.

Step 3: Find the particular solution

The particular solution is a specific solution of the non-homogeneous equation:

(d^(2)-2d+1)y=x^(2)e^(3x)

Let's use the method of undetermined coefficients to find the particular solution.

Assume the particular solution has the form:

y_p = (Ax^2 + Bx + C)e^(3x)

where A, B, and C are constants to be determined.

Step 4: Substitute into the differential equation

Substitute the particular solution into the differential equation:

(d^(2)-2d+1)(Ax^(2) + Bx + C)e^(3x) = x^(2)e^(3x)

Simplify and equate coefficients

After simplifying, we get:

9Ax^(2) + (2B - 6A)x + (C - 2B + 1) = x^(2)

Equating coefficients, we get:

9A = 1, 2B - 6A = 0, and C - 2B + 1 = 0

Solving these equations, we get:

A = 1/9, B = 1/3, and C = 2/9

Step 5: Write the general solution

The general solution is the sum of the homogeneous solution and the particular solution:

y = y_h + y_p

(c1 + c2x)e^(x) + (1/9x^(2) + 1/3x + 2/9)e^(3x)

Therefore, the general solution of the differential equation (d^(2)-2d+1)y=x^(2)e^(3x) is:

(c1 + c2x)e^(x) + (1/9x^(2) + 1/3x + 2/9)e^(3x)

where c1 and c2 are arbitrary constants.