Differential Equations
Substitution Techniques
Grade 12
Question:
<p>The solution of <span>\(\frac{dy}{dx} = y - x\)</span> is given by</p>
<p>(A) <span>\(x + C = 2(y - x) - 2\log(y - x - 1)\)</span></p>
<p>(B) <span>\(x^2 + C = (y - x)\log(y - x - 1)\)</span></p>
<p>(C) <span>\(x + C = (y - x)^2 + \log(y - x - 1)\)</span></p>
<p>(D) <span>\(y - x - 1 = Ce^{x/2}\sqrt{y - x}\)</span></p>
Step-by-Step Solution
Key Concept: Use substitution to reduce the differential equation to separable form, then apply variable separation and integration.
<p>Use substitution <span>$v = y - x$</span>, so <span>$\frac{dv}{dx} = \frac{dy}{dx} - 1 = v - 1$</span>. Separate variables and integrate to get the solution.</p>
Correct Answer: C