Differential Equations
First Order Differential Equations
Grade 12
Question:
<p>Let <span style='font-style:italic;'>y</span> = <span style='font-style:italic;'>f</span>(<span style='font-style:italic;'>x</span>) be a real valued function satisfying <span style='font-style:italic;'>x</span> \(\frac{dy}{dx} = x^2 + y - 2\), <span style='font-style:italic;'>f</span>(1) = 1, then:</p>
<p>(a) <span style='font-style:italic;'>f</span>(<span style='font-style:italic;'>x</span>) is minimum at <span style='font-style:italic;'>x</span> = 1</p>
<p>(b) <span style='font-style:italic;'>f</span>(<span style='font-style:italic;'>x</span>) is maximum at <span style='font-style:italic;'>x</span> = 1</p>
<p>(c) <span style='font-style:italic;'>f</span>(3) = 5</p>
<p>(d) <span style='font-style:italic;'>f</span>(2) = 3</p>
Step-by-Step Solution
Key Concept: Solve the linear first-order differential equation by rewriting it in standard form, then use the initial condition to find the particular solution and verify the properties at specific points.
<p><strong>Step 1: Rewrite the differential equation in standard form</strong></p><p>Given: x(dy/dx) = x² + y - 2</p><p>Divide by x: dy/dx = x + y/x - 2/x</p><p>Rearrange: dy/dx - (y/x) = x - 2/x</p><p>This is a linear ODE of the form dy/dx + P(x)y = Q(x) where P(x) = -1/x and Q(x) = x - 2/x</p></p><p><strong>Step 2: Find the integrating factor</strong></p><p>Integrating factor: μ(x) = e^∫(-1/x)dx = e^(-ln|x|) = 1/x</p></p><p><strong>Step 3: Multiply through by integrating factor and solve</strong></p><p>(1/x)(dy/dx) - (y/x²) = 1 - 2/x²</p><p>This gives: d/dx(y/x) = 1 - 2/x²</p><p>Integrate both sides: y/x = x + 2/x + C</p><p>Therefore: y = x² + 2 + Cx</p></p><p><strong>Step 4: Apply initial condition f(1) = 1</strong></p><p>1 = 1² + 2 + C(1)</p><p>1 = 3 + C</p><p>C = -2</p><p>So: f(x) = x² - 2x + 2</p></p><p><strong>Step 5: Check if f(x) has minimum at x = 1</strong></p><p>f'(x) = 2x - 2</p><p>f'(1) = 0, and f''(x) = 2 > 0</p><p>Since f'(1) = 0 and f''(1) > 0, f(x) has a minimum at x = 1 ✓ (Option a is correct)</p></p><p><strong>Step 6: Evaluate f(3) and f(2)</strong></p><p>f(3) = 9 - 6 + 2 = 5 ✓ (Option c is correct)</p><p>f(2) = 4 - 4 + 2 = 2 ≠ 3 (Option d is incorrect)</p><p><strong>∴ Answer:</strong> a, c</p>
Correct Answer: a, c