Differential Equations
Homogeneous Equations and Substitution
Grade 12
Question:
<p>The solution of <strong>x</strong> <strong>dy/dx</strong> = <strong>y</strong> + <strong>2√(y² - x²)</strong> is</p>
<p>(a) \(\frac{1}{2} \ln \left( \frac{y + \sqrt{y^2 - x^2}}{x} \right) = \ln Cx\)</p>
<p>(b) \(\ln \frac{1}{2} - x = \ln Cx\)</p>
<p>(c) \(\frac{1}{2} \ln(y - \sqrt{y^2 - x^2}) = \ln Cx\)</p>
<p>(d) \(\ln(y - \sqrt{y^2 - x^2}) = \ln Cx\)</p>
Step-by-Step Solution
Key Concept: Use the substitution y = vx for homogeneous equations of the form x(dy/dx) = f(y/x) to reduce to separable form.
<p><strong>Step 1:</strong> Recognize this as a homogeneous differential equation. Substitute $y = vx$ where $v$ is a function of $x$.</p><p><strong>Step 2:</strong> Transform the equation using the substitution and separate variables.</p><p><strong>Step 3:</strong> Integrate both sides to obtain the general solution.</p><p>∴ Answer is A.</p>
Correct Answer: A