To find the particular integral for the differential equation $(d^2 + 5d + 6)y = e^x$, we will follow the method for solving non-homogeneous linear differential equations.

Given equation:

$(d^2 + 5d + 6)y = e^x$ Here, $d$ denotes the differential operator $\frac{d}{dx}$, and $e^x$ is the non-homogeneous term.

Step 1: Characteristic Equation for the Homogeneous Part

First, solve the homogeneous part: $d^2 + 5d + 6 = 0$ Factor this quadratic equation: $(d + 2)(d + 3) = 0$ So, the roots of the characteristic equation are $d = -2$ and $d = -3$.

The complementary function (general solution to the homogeneous equation) is: $y_c = C_1 e^{-2x} + C_2 e^{-3x}$

Step 2: Particular Integral (PI)

Now, we find the particular integral. The non-homogeneous term is $e^x$. Since $e^x$ is not a solution to the homogeneous equation (the roots were -2 and -3), we try a particular solution of the form: $y_p = A e^x$ where $A$ is a constant to be determined.

Step 3: Apply the Operator

We apply the operator $d^2 + 5d + 6$ to the trial solution $y_p = A e^x$: $d(y_p) = A e^x \quad \text{and} \quad d^2(y_p) = A e^x$ So, applying the operator to $y_p$: $(d^2 + 5d + 6)y_p = A e^x + 5A e^x + 6A e^x = (1 + 5 + 6)A e^x = 12A e^x$

Step 4: Solve for $A$

Set this equal to the non-homogeneous term $e^x$: $12A e^x = e^x$ So, $A = \frac{1}{12}$.

Thus, the particular integral is: $y_p = \frac{1}{12} e^x$

Step 5: General Solution

The general solution to the differential equation is the sum of the complementary function and the particular integral: $y = y_c + y_p = C_1 e^{-2x} + C_2 e^{-3x} + \frac{1}{12} e^x$

This is the required solution.

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Would you like further details or have any questions?

Related Questions:

1. How do we handle non-homogeneous terms that involve trigonometric functions like $\sin x$ or $\cos x$?
2. What if $e^x$ was a solution to the homogeneous equation? How would the method change?
3. How do you solve second-order differential equations with repeated roots in the characteristic equation?
4. How do we find a particular integral when the non-homogeneous term is a polynomial function?
5. Can we apply variation of parameters to solve this problem instead of the method of undetermined coefficients?

Tip:

Always check if the non-homogeneous term is a solution of the homogeneous equation before assuming the form of the particular integral.