Differential Equations
Variable Separable / Direct Integration
Grade None
Question:
<p>If \(y(x)\) is the solution of the differential equation \((x+2)\frac{dy}{dx} = x^2 + 4x - 9,\ x \neq -2\) and \(y(0) = 0\), then \(y(-4)\) is equal to</p>
<p>\(0\)</p>
<p>\(1\)</p>
<p>\(-1\)</p>
<p>\(2\)</p>
Step-by-Step Solution
Key Concept: Separate the differential equation to isolate dy/dx, then integrate both sides. Use partial fraction decomposition or polynomial long division on the rational function to find the antiderivative.
<p><strong>Step 1:</strong> Separate variables: $\frac{dy}{dx} = \frac{x^2 + 4x - 9}{x+2}$</p><p><strong>Step 2:</strong> Perform polynomial long division on $\frac{x^2 + 4x - 9}{x+2}$:</p><p>$x^2 + 4x - 9 = (x+2)(x+2) + (-13) = (x+2)^2 - 13$</p><p>Therefore: $\frac{x^2 + 4x - 9}{x+2} = x + 2 - \frac{13}{x+2}$</p><p><strong>Step 3:</strong> Integrate both sides:</p><p>$y = \int \left(x + 2 - \frac{13}{x+2}\right)dx = \frac{x^2}{2} + 2x - 13\ln|x+2| + C$</p><p><strong>Step 4:</strong> Apply initial condition y(0) = 0:</p><p>$0 = 0 + 0 - 13\ln(2) + C$</p><p>$C = 13\ln(2)$</p><p><strong>Step 5:</strong> Find y(-4):</p><p>$y(-4) = \frac{16}{2} + 2(-4) - 13\ln|-2| + 13\ln(2)$</p><p>$= 8 - 8 - 13\ln(2) + 13\ln(2) = 0$</p><p>∴ Answer: A (y(-4) = 0)</p>
Correct Answer: A