Differential Equations
First-Order Differential Equations
Grade 12
Question:
<p>The solution of differential equation <span class='latex'>xdy(y^2 e^{xy} + e^{x/y}) = ydx(e^{x/y} - y^2e^{xy})</span>, is</p>
<p>(A) <span class='latex'>xy = \ln(e^x + c)</span></p>
<p>(B) <span class='latex'>x^2/y = \ln(e^{x/y} + c)</span></p>
<p>(C) <span class='latex'>xy = \ln(e^{x/y} + c)</span></p>
<p>(D) <span class='latex'>xy^2 = \ln(e^{x/y} + c)</span></p>
Step-by-Step Solution
Key Concept: Recognize that this is a homogeneous differential equation in disguise. By rearranging and grouping terms cleverly, we can identify the substitution v = x/y that simplifies the equation into a separable form.
<p><strong>Step 1: Rearrange the given equation</strong></p><p>Given: x·dy(y²e^(xy) + e^(x/y)) = y·dx(e^(x/y) - y²e^(xy))</p><p>Expanding both sides:</p><p>xy²e^(xy)dy + xe^(x/y)dy = ye^(x/y)dx - y³e^(xy)dx</p><p><strong>Step 2: Group terms strategically</strong></p><p>Rearrange to collect e^(x/y) and e^(xy) terms:</p><p>xe^(x/y)dy - ye^(x/y)dx = -y³e^(xy)dx - xy²e^(xy)dy</p><p>Factor left side: e^(x/y)(xdy - ydx) = -y²e^(xy)(ydx + xdy)</p><p><strong>Step 3: Recognize differentials</strong></p><p>Note that: d(x/y) = (ydx - xdy)/y² and d(xy) = ydx + xdy</p><p>So: xdy - ydx = -y²d(x/y) and ydx + xdy = d(xy)</p><p>Therefore: -e^(x/y)·y²d(x/y) = -y²e^(xy)d(xy)</p><p><strong>Step 4: Simplify and separate</strong></p><p>Dividing by -y²:</p><p>e^(x/y)d(x/y) = e^(xy)d(xy)</p><p><strong>Step 5: Integrate both sides</strong></p><p>∫e^(x/y)d(x/y) = ∫e^(xy)d(xy)</p><p>e^(x/y) = e^(xy) + c</p><p><strong>Step 6: Verify with the substitution xy = t</strong></p><p>If xy = t, then e^(x/y) = e^(xy) + c becomes:</p><p>e^(x/y) = e^(xy) + c, which rearranges to xy = ln(e^(x/y) + c)</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C