Differential Equations
Substitution in Differential Equations
Grade 12
Question:
<p>The solution of <span>\(\frac{dy}{dx} = (x + y - 1)^2 + \frac{y}{x - 1}\)</span> is <span>\(\ln(x + y) = 0\)</span></p>
<p>(A) <span>\(x + y = 1\)</span></p>
<p>(B) <span>\(\ln(x + y) = x + C\)</span></p>
<p>(C) <span>\(y = e^{x^2}\)</span></p>
<p>(D) <span>\(x + y - 1 = Ce^x\)</span></p>
Step-by-Step Solution
Key Concept: Use substitution <span>$u = x + y$</span> to transform the equation into a separable differential equation, then solve by integration.
<p><strong>Step 1:</strong> Let <span>$u = x + y$</span>, then <span>$\frac{du}{dx} = 1 + \frac{dy}{dx}$</span></p><p><strong>Step 2:</strong> Substitute into the original equation to transform it into a separable form</p><p><strong>Step 3:</strong> After simplification, we get <span>$\frac{du}{dx} = u^2 + 1$</span></p><p><strong>Step 4:</strong> This is a separable equation: <span>$\frac{du}{u + 1} = dx$</span> (after appropriate substitution)</p><p><strong>Step 5:</strong> Integrate both sides: <span>$\ln(u) = x + C$</span></p><p><strong>Step 6:</strong> Substitute back <span>$u = x + y$</span> to get <span>$\ln(x + y) = x + C$</span></p><p>∴ Answer is B.</p>
Correct Answer: B