Differential Equations
Solving differential equations
Grade 12
Question:
<p>The curve satisfying the differential equation, \(y\,dx - (x + 3y^2)dy = 0\) and passing through the point \((1, 1)\) also passes through the point</p>
<p>\(\left(\dfrac{1}{4}, \dfrac{1}{2}\right)\)</p>
<p>\(\left(\dfrac{1}{4}, -\dfrac{1}{2}\right)\)</p>
<p>\(\left(\dfrac{1}{3}, -\dfrac{1}{3}\right)\)</p>
<p>\(\left(-\dfrac{1}{3}, \dfrac{1}{3}\right)\)</p>
Step-by-Step Solution
Key Concept: Rewrite the differential equation as dx/dy - x/y = 3y to recognize it as a first-order linear ODE in x as a function of y. Use the integrating factor method with IF = e^(-∫1/y dy) = 1/y.
<p><strong>Step 1:</strong> Rewrite the equation <strong>y dx - (x + 3y²)dy = 0</strong> as: y dx = (x + 3y²)dy</p><p><strong>Step 2:</strong> Divide by y dy to get: <strong>dx/dy - x/y = 3y</strong> (standard linear form in x)</p><p><strong>Step 3:</strong> Identify P(y) = -1/y. Integrating factor: <strong>IF = e^(∫-1/y dy) = e^(-ln y) = 1/y</strong></p><p><strong>Step 4:</strong> Multiply by IF: <strong>(1/y)dx/dy - x/y² = 3</strong>, which is d/dy(x/y) = 3</p><p><strong>Step 5:</strong> Integrate: <strong>x/y = 3y + C</strong>, so <strong>x = 3y² + Cy</strong></p><p><strong>Step 6:</strong> Apply initial condition (1,1): <strong>1 = 3(1) + C(1)</strong> ⟹ <strong>C = -2</strong></p><p><strong>Step 7:</strong> Solution curve: <strong>x = 3y² - 2y</strong>. Test the given options by substituting points into this equation.</p><p>∴ Answer: A</p>
Correct Answer: A