Statement A: Exact Differential Equation Check

The equation $(1 + e^x y + x e^x y) dx + (xe^x + 2) dy = 0$ is of the form $M dx + N dy = 0$. Here, $M = 1 + e^x y + x e^x y$ and $N = xe^x + 2$. For the equation to be exact, $\frac{\partial M}{\partial y}$ must equal $\frac{\partial N}{\partial x}$.

Calculating the partial derivatives:

• $\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (1 + y(e^x + x e^x)) = e^x + x e^x$
• $\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (xe^x + 2) = (1 \cdot e^x + x \cdot e^x) + 0 = e^x + x e^x$

Since $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the differential equation is exact. Statement A is True.

Statement B: Particular Solution for $(D^2 - D - 2)y = e^{-x}$

The auxiliary equation is $m^2 - m - 2 = 0 \implies (m-2)(m+1)=0$, with roots $m=2, m=-1$. The complementary function is $y_c = c_1 e^{2x} + c_2 e^{-x}$.

Since $e^{-x}$ is part of $y_c$, the particular solution ($y_p$) guess is $y_p = Ax e^{-x}$. Derivatives:

• $Dy_p = A(e^{-x} - x e^{-x})$
• $D^2y_p = A(-2e^{-x} + x e^{-x})$

Substituting into $(D^2 - D - 2)y = e^{-x}$:

$ A(-2e^{-x} + x e^{-x}) - A(e^{-x} - x e^{-x}) - 2(A x e^{-x}) = e^{-x} $

$ A e^{-x} (-2 + x - 1 + x - 2x) = e^{-x} $

$ -3A e^{-x} = e^{-x} \implies A = -\frac{1}{3} $

The particular solution is $y_p = -\frac{1}{3} x e^{-x}$. Statement B is True.

Statement C: Particular Solution for $(D^2 + 4)y = \sin^2 x$

Convert the right-hand side: $\sin^2 x = \frac{1 - \cos(2x)}{2} = \frac{1}{2} - \frac{1}{2} \cos(2x)$. The equation becomes $(D^2 + 4)y = \frac{1}{2} - \frac{1}{2} \cos(2x)$.

Find the particular solution ($y_p$):

• For the term $\frac{1}{2}$: $y_{p1} = K$. Then $(D^2+4)K = 0+4K = \frac{1}{2} \implies K = \frac{1}{8}$.
• For the term $-\frac{1}{2} \cos(2x)$: Using the rule $\frac{1}{D^2+a^2} \cos(ax) = \frac{x}{2a} \sin(ax)$ with $a=2$. $y_{p2} = \frac{1}{D^2+4} (-\frac{1}{2} \cos(2x)) = -\frac{1}{2} \left( \frac{x}{2(2)} \sin(2x) \right) = -\frac{x}{8} \sin(2x)$.

The full particular solution is $y_p = y_{p1} + y_{p2} = \frac{1}{8} - \frac{x}{8} \sin(2x)$.

The statement claims the particular solution is only $-\frac{x}{8} \sin(2x)$, which is incomplete as it omits the constant term $\frac{1}{8}$. Statement C is False.

Statement D: Linear Independence of $x^2$ and $x|x|$

Check if $c_1 x^2 + c_2 x |x| = 0$ for all $x \in (-\infty, \infty)$ implies $c_1 = c_2 = 0$. For $x > 0$, $|x| = x$, so $c_1 x^2 + c_2 x^2 = 0 \implies (c_1 + c_2)x^2 = 0 \implies c_1 + c_2 = 0$. For $x < 0$, $|x| = -x$, so $c_1 x^2 + c_2 x (-x) = 0 \implies (c_1 - c_2)x^2 = 0 \implies c_1 - c_2 = 0$. Solving the system $c_1 + c_2 = 0$ and $c_1 - c_2 = 0$ yields $c_1 = 0$ and $c_2 = 0$.

Thus, the functions are linearly independent. Statement D is True.

Conclusion on Statements

Based on the analysis:

• Statement A: True
• Statement B: True
• Statement C: False
• Statement D: True

The correct combination of true statements is A, B, and D.