Differential Equations
Linear Differential Equations
Grade 12
Question:
<p>A curve passes through the point (1, 1) and satisfies the differential equation
\(\frac{dx}{dy} - \frac{1}{y}x = 3y\). Which of the following points lies on the curve?</p>
<p>\(\left(\frac{-1}{3}, \frac{1}{3}\right)\)</p>
<p>\(\left(\frac{1}{3}, \frac{-1}{3}\right)\)</p>
<p>\(\left(\frac{-1}{3}, \frac{1}{3}\right)\)</p>
<p>\(\left(\frac{1}{3}, \frac{1}{3}\right)\)</p>
Step-by-Step Solution
Key Concept: Recognize this as a linear first-order differential equation in x as a function of y. Rearrange to standard form dx/dy - (1/y)x = 3y and use integrating factor method with μ(y) = e^(-∫(1/y)dy) = e^(-ln y) = 1/y.
<p><strong>Step 1:</strong> Recognize the standard form: dx/dy - (1/y)x = 3y, where x depends on y.</p><p><strong>Step 2:</strong> Find integrating factor: μ(y) = e^(∫(-1/y)dy) = e^(-ln y) = 1/y.</p><p><strong>Step 3:</strong> Multiply entire equation by μ(y) = 1/y: (1/y)(dx/dy) - (1/y²)x = 3.</p><p><strong>Step 4:</strong> Recognize left side as d/dy[x/y]: d/dy[x/y] = 3.</p><p><strong>Step 5:</strong> Integrate both sides: x/y = 3y + C.</p><p><strong>Step 6:</strong> Apply initial condition (1,1): 1/1 = 3(1) + C → 1 = 3 + C → C = -2.</p><p><strong>Step 7:</strong> Solution: x/y = 3y - 2, or x = 3y² - 2y.</p><p><strong>Step 8:</strong> Verify and test given options by substituting into x = 3y² - 2y.</p><p>∴ Answer: A</p>
Correct Answer: A