Question

A particular integral of the differential equation $\frac{d^2 y}{dx^2} + 3 \frac{dy}{dx} + 4y = 3x + 2$ is given by}

• A. $y_p(x) = -\frac{3}{4}x - \frac{1}{16}$
• B. $y_p(x) = \frac{1}{16}x - \frac{3}{4}$
• C. $y_p(x) = -\frac{3}{4}x + \frac{1}{16}$
• D. $y_p(x) = \frac{3}{4}x - \frac{1}{16}$

Hint:

To test the options quickly, differentiate each candidate $y_p(x)$ and substitute it back into the left-hand side of the differential equation to see which one evaluates to $3x + 2$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The particular integral ($y_p$) of a non-homogeneous linear differential equation is a specific solution that satisfies the non-zero right-hand side function.
Key Formula or Approach:
For a linear right-hand side $f(x) = 3x + 2$, we use the method of undetermined coefficients and assume a trial solution of the same polynomial form:
\[ y_p(x) = Ax + B \]

Step 2: Detailed Explanation:
Let the trial particular solution be:
\[ y_p(x) = Ax + B \]
Calculate the first and second derivatives of $y_p$:
\[ y_p'(x) = A \]
\[ y_p''(x) = 0 \]
Substitute these derivatives into the given differential equation:
\[ 0 + 3(A) + 4(Ax + B) = 3x + 2 \]
Rearrange the left-hand side:
\[ 4Ax + (3A + 4B) = 3x + 2 \]
Compare the coefficients of like terms on both sides of the equation:
1. Compare coefficients of $x$:
\[ 4A = 3 \implies A = \frac{3}{4} \]
2. Compare the constant terms:
\[ 3A + 4B = 2 \]
Substitute $A = \frac{3}{4}$:
\[ 3\left(\frac{3}{4}\right) + 4B = 2 \]
\[ \frac{9}{4} + 4B = 2 \]
\[ 4B = 2 - \frac{9}{4} = -\frac{1}{4} \implies B = -\frac{1}{16} \]
Substitute $A$ and $B$ back into the trial solution:
\[ y_p(x) = \frac{3}{4}x - \frac{1}{16} \]
Therefore, the particular integral is $\frac{3}{4}x - \frac{1}{16}$.

Step 3: Final Answer
The correct option is (D).