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GATE 2022 ME (ME2) – Question 36

Engineering Mathematics · Differential Equations: First order and higher order linear equations, Euler-Cauchy equation · 2 marks · Multiple choice

For the exact differential equation $\frac{du}{dx}=\frac{-xu^2}{2+x^2u}$, which one of the following is the solution?

  1. $u^2+2x^2=\text{constant}$
  2. $xu^2+u=\text{constant}$
  3. $\frac12x^2u^2+2u=\text{constant}$
  4. $\frac12ux^2+2x=\text{constant}$

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Correct answer: (C) $\frac12x^2u^2+2u=\text{constant}$

Explanation

The equation is
$$\frac{du}{dx}=\frac{-xu^2}{2+x^2u}\;\Longleftrightarrow\;xu^2\,dx+(2+x^2u)\,du=0 .$$

**Check exactness.** With $M=xu^2$ and $N=2+x^2u$:
$$\frac{\partial M}{\partial u}=2xu=\frac{\partial N}{\partial x},$$
so the equation is exact.

**Find the potential $\phi$.**
$$\phi=\int M\,dx=\frac{x^2u^2}{2}+g(u),\qquad \frac{\partial\phi}{\partial u}=x^2u+g^\prime(u)=2+x^2u\;\Rightarrow\;g(u)=2u .$$

So the solution is $\tfrac12x^2u^2+2u=\text{constant}$ (option **C**).

**Check by differentiation:** $\dfrac{d}{dx}\left(\tfrac12x^2u^2+2u\right)=xu^2+(x^2u+2)u^\prime=0$, which is the given equation.