To solve the differential equation $y^{\prime\prime} + y = x^2 + 3 + \cos x$, we need to find the general solution. This equation is non-homogeneous, so the general solution is composed of two parts: the complementary solution (solving the homogeneous equation) and the particular solution (accounting for the non-homogeneous part).

Step 1: Solve the Homogeneous Equation

The homogeneous equation is: $y^{\prime\prime} + y = 0$ The characteristic equation for this is: $r^2 + 1 = 0$ Solving for $r$: $r = \pm i$ Thus, the complementary solution is: $y_c = c_1 \cos x + c_2 \sin x$ where $c_1$ and $c_2$ are constants.

Step 2: Find a Particular Solution

We now find the particular solution for the non-homogeneous equation: $y^{\prime\prime} + y = x^2 + 3 + \cos x$ We break the right-hand side into two parts:

1. $x^2 + 3$
2. $\cos x$

Particular Solution for $x^2 + 3$

Assume a particular solution of the form: $y_p^{(1)} = Ax^2 + Bx + C$ Substituting this into the equation $y^{\prime\prime} + y = x^2 + 3$, we find the coefficients $A$, $B$, and $C$.

The second derivative of $y_p^{(1)}$ is: $y_p^{(1)\prime\prime} = 2A$ Substitute $y_p^{(1)}$ and $y_p^{(1)\prime\prime}$ into the equation: $2A + (Ax^2 + Bx + C) = x^2 + 3$ Equating coefficients of like powers of $x$, we get: $A = 1, \quad B = 0, \quad 2A + C = 3 \implies C = 1$ So, the particular solution for this part is: $y_p^{(1)} = x^2 + 1$

Particular Solution for $\cos x$

Since $\cos x$ is already a solution to the homogeneous equation, we assume a particular solution of the form: $y_p^{(2)} = x(A \cos x + B \sin x)$ Substitute this into $y^{\prime\prime} + y = \cos x$, and solve for $A$ and $B$.

The second derivative of $y_p^{(2)}$ is more involved, but after solving, we find: $y_p^{(2)} = -\frac{1}{2}x \sin x$

Step 3: General Solution

The general solution is the sum of the complementary and particular solutions: $y = y_c + y_p = (c_1 \cos x + c_2 \sin x) + (x^2 + 1) - \frac{1}{2}x \sin x$

Thus, the general solution is: $y = c_1 \cos x + \left( c_2 - \frac{1}{2}x \right) \sin x + x^2 + 1$

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Would you like further details or have any questions?

Here are five related questions:

1. How do you find the complementary solution of a second-order differential equation?
2. What is the method of undetermined coefficients in solving differential equations?
3. Why is it necessary to modify the particular solution for $\cos x$?
4. Can you explain how to handle the second derivative of a product like $x \cos x$?
5. How would the solution change if the right-hand side were $e^x$ instead of $\cos x$?

Tip: Always check if your particular solution is already part of the complementary solution to avoid redundancy.