Question

Consider the following differential equation:
\[ t^2\dfrac{d^2y}{dt^2}+7t\dfrac{dy}{dt}+8ty=10\sin(t) \]
Which one of the following options is correct?

• A. It is a linear differential equation
• B. It is a nonlinear differential equation
• C. It is a time-invariant differential equation
• D. It is a second-order partial differential equation

Hint:

Check if y and its derivatives appear only to the first power for linearity, separately from whether the coefficients depend on t for time invariance.

Answer 1

The Correct Option is A

Step 1: Check whether the equation is ordinary or partial.
The unknown function \(y\) appears only as \(y(t)\), a function of the single independent variable \(t\), and only ordinary derivatives \(d^2y/dt^2\) and \(dy/dt\) appear, never partial derivatives with respect to two or more variables. So this is an ordinary differential equation, not a partial differential equation, which rules out option (D).
Step 2: Check linearity.
A differential equation in \(y\) is linear when \(y\) and all of its derivatives appear only to the first power, are never multiplied together, and never pass through a nonlinear function such as \(y^2\), \(\sin(y)\), or \(y\cdot y'\); the coefficients multiplying \(y\), \(y'\), \(y''\) are allowed to depend on the independent variable \(t\). Here, \(y''\), \(y'\), and \(y\) each appear to the first power, multiplied only by coefficients \(t^2\), \(7t\), \(8t\) that depend on \(t\), and the right-hand side \(10\sin(t)\) is a known function of \(t\) only, not of \(y\). So the equation is linear. This rules out option (B) and confirms option (A).
Step 3: Check time invariance.
A linear differential equation is time-invariant only when every coefficient multiplying \(y\) and its derivatives is a CONSTANT, not depending on \(t\). Here the coefficients \(t^2\), \(7t\), and \(8t\) explicitly depend on \(t\), so this equation is linear but time-VARYING, not time-invariant. This rules out option (C). This particular equation is in fact a form of the Euler-Cauchy (equidimensional) equation, a classic example of a linear, time-varying ODE.
Step 4: Final Answer.
Since the equation is linear in \(y\) and its derivatives, option (A) is the correct statement.
\[ \boxed{\text{Linear differential equation}} \]