Question

Given a differential equation \(\dfrac{d^2x}{dt^2} + x= 0\) with \(x(0) \neq 0\).
Which of the following statements is/are true?

• A. \(|x(t)|^2 + \left|\dfrac{dx(t)}{dt}\right|^2= 0\) for all \(t\geq0\)
• B. \(|x(t)|^2 + \left|\dfrac{dx(t)}{dt}\right|^2= c\) for all \(t\geq0\), for some real constant \(c>0\)
• C. \(|x(t)|^2 + \left|\dfrac{dx(t)}{dt}\right|^2= e^{jt}\) for all \(t\geq0\)
• D. \(|x(t)|^2 + \left|\dfrac{dx(t)}{dt}\right|^2= \sin t\) for all \(t\geq0\)

Hint:

Solve as \(x(t)=C_1\cos t + C_2\sin t\), or note \(\frac{d}{dt}(x^2+(x')^2)=2x'(x+x'')=0\) since \(x''=-x\); the sum is a constant fixed by \(x(0)\neq0\).

Answer 1

The Correct Option is B

Step 1: Look for a conserved quantity without solving the equation first.
Instead of writing down $x(t)$ explicitly, define $V(t)=x(t)^2+\left(\dfrac{dx}{dt}\right)^2$ and check how it changes with time directly from the differential equation $x''+x=0$.
Step 2: Differentiate V(t) with respect to t.
\[ \frac{dV}{dt} = 2x\frac{dx}{dt} + 2\frac{dx}{dt}\frac{d^2x}{dt^2} = 2\frac{dx}{dt}\left(x+\frac{d^2x}{dt^2}\right) \]
Step 3: Use the differential equation to kill the bracket.
The given equation says $x+\dfrac{d^2x}{dt^2}=0$ for every $t$ (that is exactly what $x''+x=0$ means). Substituting this in,
\[ \frac{dV}{dt} = 2\frac{dx}{dt}\times0 = 0 \quad\text{for all }t \]
Step 4: Conclude V(t) is constant.
A quantity whose derivative is zero everywhere does not change with time, so $V(t)=V(0)$ for all $t\geq0$. This means $|x(t)|^2+\left|\dfrac{dx}{dt}\right|^2$ equals a fixed number $c=V(0)$, valid for every $t\geq0$, matching the shape of statement (B).
Step 5: Show c is positive using the initial condition.
\[ c = V(0) = x(0)^2 + x'(0)^2 \]
We are told $x(0)\neq0$, so $x(0)^2>0$, and $x'(0)^2\geq0$ always, so $c>0$ strictly.
Step 6: Compare against each option.
Statement (A) needs $c=0$, impossible since $c>0$. Statement (C) needs a complex, time-varying value $e^{jt}$, but $V(t)$ is manifestly a sum of two real squares, always real and constant, so it can equal $e^{jt}$ at most at the single instant $t=0$, not for all $t\geq0$. Statement (D) needs $V(t)=\sin t$, which changes sign and revisits $0$, contradicting that $V$ is a fixed positive number. Only statement (B) survives this check.
\[ \boxed{|x(t)|^2+\left|\tfrac{dx}{dt}\right|^2=c,\ \text{a fixed positive real constant}} \]