Question

Among the following, the differential equation(s) not representative of a linear dynamical system is/are ____.
(i) \[ \frac{d^2x}{dt^2} + 2t\frac{dx}{dt} + x(t) = u(t) \]
(ii) \[ \left(\frac{dx}{dt}\right)^2 + 2\frac{dx}{dt} + x(t) = u(t) \]
(iii) \[ \frac{d^2x}{dt^2} + 2\frac{dx}{dt} + x(t) = u(t) \]

• A. (i) and (ii) only
• B. (i) only
• C. (ii) only
• D. (ii) and (iii) only

Hint:

Linearity only cares about whether \(x\), \(u\), and their derivatives appear to the first power with no products between them; time-varying coefficients alone do not break linearity.

Answer 1

The Correct Option is C

Step 1: Understanding the Question.
We are given three differential equations that relate an input \(u(t)\) to a variable \(x(t)\), and need to say which of them do NOT represent a linear dynamical system. A system is linear if it obeys the superposition principle, additivity and homogeneity, which in terms of the differential equation means every term must be a constant (or time-varying) coefficient multiplied by \(x\), \(u\), or one of their derivatives, raised only to the first power, with no products or powers of \(x\), \(u\), or their derivatives with each other.

Step 2: Key Formula or Approach.
A linear differential equation has the general form
\[ a_n(t)\frac{d^n x}{dt^n} + \cdots + a_1(t)\frac{dx}{dt} + a_0(t) x(t) = b(t)\, u(t) \]
where the coefficients \(a_i(t)\) are allowed to depend on time (this only affects time-invariance, not linearity), but \(x\), \(u\), and their derivatives themselves must appear only to the first power and never multiplied together. If any term breaks this, for example a squared derivative, a product of \(x\) and \(u\), or a function like \(\sin(x)\), the equation is nonlinear.

Step 3: Detailed Explanation.
Equation (i): \(\frac{d^2x}{dt^2} + 2t\frac{dx}{dt} + x(t) = u(t)\). Every term is \(x\), \(u\), or a derivative of \(x\), each appearing to the first power, multiplied only by coefficients that depend on \(t\) (the \(2t\) in front of \(\frac{dx}{dt}\)). A time-varying coefficient does not break linearity, it only makes the system time-varying rather than time-invariant. So equation (i) IS linear (linear time-varying).
Equation (ii): \(\left(\frac{dx}{dt}\right)^2 + 2\frac{dx}{dt} + x(t) = u(t)\). The first term is \(\left(\frac{dx}{dt}\right)^2\), the derivative of \(x\) raised to the second power. Superposition fails here: if \(x_1\) and \(x_2\) each produce their own response, doubling \(x\) does not double this squared term, it quadruples it. This is a genuinely nonlinear term, so equation (ii) is NOT linear.
Equation (iii): \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + x(t) = u(t)\). Every term is \(x\), \(u\), or a derivative of \(x\) to the first power, with constant coefficients. This is linear, and since the coefficients do not even depend on \(t\), it is linear time-invariant as well.

Step 4: Final Answer.
Only equation (ii) contains a nonlinear term (the squared derivative), while (i) and (iii) are both linear (with (i) being time-varying and (iii) being time-invariant). So the set of equations that are NOT representative of a linear dynamical system is just (ii). Option (A) wrongly includes (i), which is linear despite its time-varying coefficient. Option (B) picks only (i), which is actually linear, not the nonlinear one. Option (D) wrongly includes (iii), which has constant coefficients and no nonlinear terms.
\[ \boxed{\text{(ii) only}} \]