Solve the given differential equation by undetermined coefficients: $y'' + 6y' + 9y = -xe^{9x}$

Neetesh Kumar | November 05, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh4Sec05 (Homework) Question - 9
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Step-by-Step-Solution:

To solve this differential equation, we need to find both the complementary solution $y_c(x)$ and the particular solution $y_p(x)$.

Find the Complementary Solution, $y_c(x)$:

The complementary equation is:

$y'' + 6y' + 9y = 0$

Rewrite this as the characteristic equation:

$r^2 + 6r + 9 = 0$

This factors as:

$(r + 3)^2 = 0$

So we have a repeated root at $r = -3$. Thus, the complementary solution is:

$y_c(x) = (C_1 + C_2 x)e^{-3x}$

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Find the Particular Solution, $y_p(x)$:

The non-homogeneous term is $-xe^{9x}$. Since $e^{9x}$ is not part of the complementary solution, we try a particular solution of the form:

$y_p(x) = (Ax + B) e^{9x} = Ax e^{9x} + B e^{9x}$

Differentiate $y_p(x)$ to find $y_p'$ and $y_p''$:

• First derivative $y_p'$:

  $y_p' = (9Ax + A + 9B)e^{9x}$

• Second derivative $y_p''$:

  $y_p' = (81Ax+18A+81B)e^{9x}$

Putting these values in the original equation:

$(81Ax+18A+81B)e^{9x} + (54Ax + 6A + 54B)e^{9x} + (9Ax + 9B) e^{9x} = -xe^{9x}$

After simplifying the terms:

$(144Ax + 24A + 144B)e^{9x} = -xe^{9x}$

After comparing terms on both sides:

$A = -\frac{1}{144}, B = \frac{1}{864}$

So we can write the particular solution as:

$y_p(x) = (-\frac{1}{144}x + \frac{1}{864}) e^{9x}$

So the general solution of the above DE:

$y = y_c(x) + y_p(x)$

Final Answer:

$y = \boxed{(C_1 + C_2 x)e^{-3x} + (-\frac{1}{144}x + \frac{1}{864}) e^{9x}}$

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