Solve the given differential equation by undetermined coefficients: $y'' + 9y = 7 \sin(x)$

Neetesh Kumar | November 05, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh4Sec05 (Homework) Question - 8
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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:

$y'' + 9y = 0$

Rewrite this equation using the characteristic equation:

$r^2 + 9 = 0$

Solving for $r$, we get:

$r = \pm 3i$

This gives us complex roots, so the complementary solution is:

$y_c(x) = C_1 \cos(3x) + C_2 \sin(3x)$

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

Since the non-homogeneous term is $7 \sin(x)$, we assume a particular solution of the form:

$y_p(x) = A \cos(x) + B \sin(x)$

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

• $y_p' = -A \sin(x) + B \cos(x)$
• $y_p'' = -A \cos(x) - B \sin(x)$

Substitute $y_p$ and $y_p''$ into the original differential equation:

$y'' + 9y = 7 \sin(x)$

This becomes:

$(-A \cos(x) - B \sin(x)) + 9(A \cos(x) + B \sin(x)) = 7 \sin(x)$

Simplify this equation:

$(9A - A) \cos(x) + (9B - B) \sin(x) = 7 \sin(x)$

Which simplifies further to:

$8A \cos(x) + 8B \sin(x) = 7 \sin(x)$

Equate the coefficients of $\cos(x)$ and $\sin(x)$:

• For $\cos(x)$: $8A = 0 \Rightarrow A = 0$
• For $\sin(x)$: $8B = 7 \Rightarrow B = \frac{7}{8}$

Therefore, the particular solution is:

$y_p(x) = \frac{7}{8} \sin(x)$

Combine the Solutions:

The general solution is the sum of the complementary and particular solutions:

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

$y(x) = C_1 \cos(3x) + C_2 \sin(3x) + \frac{7}{8} \sin(x)$

Final Answer

$y(x) = \boxed{C_1 \cos(3x) + C_2 \sin(3x) + \frac{7}{8} \sin(x)}$

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