Solve the following differential equation : (𝐷^2 − 4𝐷 + 4)𝑦 = 𝑥^2𝑒^3𝑥 + sin^2 𝑥

To solve the differential equation \((D^2 - 4D + 4)y = x^2 e^{3x} + \sin^2 x\), we need to break it down into two parts: finding the complementary solution (homogeneous part) and the particular solution (non-homogeneous part).

Step 1: Solve the Homogeneous Equation

The first step is to solve the homogeneous equation associated with the differential operator:

(D^2 - 4D + 4)y = 0

This can be factored as:

(D - 2)^2 y = 0

From this, we can determine the characteristic equation:

r^2 - 4r + 4 = 0

Factoring gives us:

(r - 2)^2 = 0

This has a double root at \(r = 2\). Therefore, the general solution to the homogeneous equation is:

y_h = C_1 e^{2x} + C_2 x e^{2x}

where \(C_1\) and \(C_2\) are constants determined by initial conditions.

Step 2: Find the Particular Solution

Next, we need to find a particular solution \(y_p\) for the non-homogeneous equation:

(D^2 - 4D + 4)y = x^2 e^{3x} + \sin^2 x

We will tackle each term on the right-hand side separately.

Particular Solution for \(x^2 e^{3x}\)

For the term \(x^2 e^{3x}\), we can use the method of undetermined coefficients. Since \(e^{3x}\) is not a solution of the homogeneous equation, we can propose a particular solution of the form:

y_{p1} = A x^2 e^{3x} + B x e^{3x} + C e^{3x}

We will differentiate this expression and substitute it back into the left-hand side of the original equation to solve for \(A\), \(B\), and \(C\).

Particular Solution for \(\sin^2 x\)

Next, we need to find a particular solution for \(\sin^2 x\). We can use the identity:

\(\sin^2 x = \frac{1 - \cos(2x)}{2}\)

Thus, we can consider a particular solution of the form:

y_{p2} = D + E \cos(2x) + F \sin(2x)

Again, we will differentiate this expression and substitute it back into the left-hand side of the original equation to solve for \(D\), \(E\), and \(F\).

Step 3: Combine Solutions

Once we have both particular solutions \(y_{p1}\) and \(y_{p2}\), we can combine them with the homogeneous solution:

y = y_h + y_{p1} + y_{p2}

Step 4: Final Solution

Finally, the complete solution to the differential equation will be:

y = C_1 e^{2x} + C_2 x e^{2x} + y_{p1} + y_{p2}

To find the specific values of \(C_1\) and \(C_2\), you would need initial or boundary conditions provided in the problem.

In summary, solving this differential equation involves finding both the complementary and particular solutions, which together give us the complete solution to the equation. If you have any specific values or conditions, we can further refine the solution!