Differential Equations
Exact Differential Equations
Grade 12
Question:
<p>The solution of \((y(1 + x^{-1}) + \sin y) dx + (x + \ln x + x \cos y) dy = 0\) is</p>
<p>(A) \((1 + y^{-1} \sin y) + x^{-1} \ln x = C\)</p>
<p>(B) \((y + \sin y) + xy \ln x = C\)</p>
<p>(C) \(xy + y \ln x + x \sin y = C\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Check if the differential equation is exact by verifying that partial derivatives are equal, then integrate to find the solution.
<p>The given equation is $(y(1 + x^{-1}) + \sin y) dx + (x + \ln x + x \cos y) dy = 0$</p><p>Rewrite as: $(y + \frac{y}{x} + \sin y) dx + (x + \ln x + x \cos y) dy = 0$</p><p>Check if it's exact: $\frac{\partial M}{\partial y} = 1 + \frac{1}{x} + \cos y = \frac{\partial N}{\partial x}$</p><p>It is exact. The solution is $F(x,y) = C$ where $\frac{\partial F}{\partial x} = M$ and $\frac{\partial F}{\partial y} = N$</p><p>Integrating: $F = xy + y\ln x + x\sin y = C$</p>
Correct Answer: C