The GATE Grind

GATE 2024 EE – Question 45

Engineering Mathematics · Differential Equations: First order equations, Higher order linear differential equations, Variation of parameters, Cauchy and Euler equations, Partial differential equations, Separation of variables · 2 marks · Multiple select

Which of the following differential equations is/are nonlinear?

  1. $t\,x(t)+\dfrac{dx(t)}{dt}=t^2e^t,\quad x(0)=0$
  2. $\dfrac12e^t+x(t)\dfrac{dx(t)}{dt}=0,\quad x(0)=0$
  3. $x(t)\cos t-\dfrac{dx(t)}{dt}\sin t=1,\quad x(0)=0$
  4. $x(t)+e^{\left(\frac{dx(t)}{dt}\right)}=1,\quad x(0)=0$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $\dfrac12e^t+x(t)\dfrac{dx(t)}{dt}=0,\quad x(0)=0$; (D) $x(t)+e^{\left(\frac{dx(t)}{dt}\right)}=1,\quad x(0)=0$

Explanation

(A) and (C) are linear in $x$ and $x'$ (with $t$-dependent coefficients). (B) contains the product $x\,x'$, and (D) contains $e^{x'}$, a nonlinear function of the derivative. So B and D are nonlinear.