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GATE 2026 IN – Question 54

Engineering Mathematics · Differential equations · 2 marks · Multiple select

Given the differential equation $d^2x/dt^2+x=0$ with $x(0)\ne0$, which of the following statements is/are true?

  1. $|x(t)|^2+\left|\frac{dx(t)}{dt}\right|^2=0$ for all $t\ge0$.
  2. $|x(t)|^2+\left|\frac{dx(t)}{dt}\right|^2=c$ for all $t\ge0$, for some real constant $c>0$.
  3. $|x(t)|^2+\left|\frac{dx(t)}{dt}\right|^2=e^{jt}$ for all $t\ge0$.
  4. $|x(t)|^2+\left|\frac{dx(t)}{dt}\right|^2=\sin t$ for all $t\ge0$.

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Correct answer: (B) $|x(t)|^2+\left|\frac{dx(t)}{dt}\right|^2=c$ for all $t\ge0$, for some real constant $c>0$.

Explanation

For a real solution, let $E(t)=x(t)^2+\dot x(t)^2$. Differentiation gives $\dot E=2x\dot x+2\dot x\ddot x=2\dot x(x+\ddot x)=0$. Hence $E$ is constant.

Its initial value is $E(0)=x(0)^2+\dot x(0)^2>0$ because $x(0)\ne0$. Thus **B** is true. For complex solutions, differentiating the squared magnitudes using conjugates gives the same constant result.