List of practice Questions

If the Laplace transform of \(f(t)\) and \(f'(t)\) exist and \(F(s)\) is the Laplace transform of \(f(t)\), then \[ \lim_{s\to\infty}sF(s) \] is
Consider differential equation \(\dfrac{dy}{dx}+xy=x\) with the condition as \(y=0\) at \(x=0\). The value of \(y\) at \(x=1.0\) is ______ (rounded off to two decimal places).
Consider the differential equation \(\dot{\vec{w}}=A\vec{w}\), with \(\vec{w}(t=0)=\begin{bmatrix}1\\1\end{bmatrix}\). If \(\vec{w}(t)=e^{t}\vec{u}_x+e^{-2t}\vec{u}_y\) is the solution to the equation, where \(\vec{u}_x\) and \(\vec{u}_y\) are unit vectors along the positive \(x\) and \(y\) axes respectively, then which of the following options is the correct matrix representing \(A\)?
Consider the second-order differential equation
\[ \frac{d^2y}{dx^2}+\frac{dy}{dx}+y=0 \]
with initial conditions
\[ y(0)=1,\quad \left.\frac{dy}{dx}\right|_{x=0}=1 \]
The solution is given by
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?
An ordinary differential equation is given below.
\[ x^2\frac{d^2y}{dx^2} = 6y \]
Considering \(a\) and \(b\) as arbitrary constants, the general solution of the equation is
The equation \(\dfrac{d^2y}{dx^2} - y = 0\) has a solution of the form \(y = e^{Ax}\). The value(s) of \(A\) satisfying this is/are:
Consider the differential equation with the initial conditions given below. If \(y(x)\) is the solution of the equation, the value of the slope, \(\frac{dy}{dx}\), at \(x=\ln(2)\) is ________ (rounded off to three decimal places).
\[ \frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=0 \quad \text{with} \quad y|_{x=0}=0 \text{ and } \left.\frac{dy}{dx}\right|_{x=0}=1 \]
An object of mass 1 kg is launched with an initial speed of \(v_o\) into a large tank filled with a viscous liquid. The liquid exerts a resistive force (drag) of the form \(D = \alpha v\) on any object that is moving inside it, where \(v\) is the instantaneous speed of the object and \(\alpha = 1\) kg/s. If the effect of gravity is ignored, the time taken by the object to slow down to the speed \(v_o/2\) is ________ s (rounded off to 2 decimal places). Assume that the tank is sufficiently large for the above deceleration to happen inside the tank.
The initial value problem \[ (x-x^2)\frac{dy}{dx} = (2x-1)y, \qquad y(x_0)=y_0, \] has a unique solution if \((x_0,y_0)\) equals to
The particular integral of the differential equation \[ (D^2+1)y = \sin x\,\sin2x \] is
The general solution of \[ \log\left(\frac{dy}{dx}\right)=ax+by \] is
If \(y_1,\;y_2\) are two dependent solutions of a second order linear homogeneous differential equation then \(y_1y_2'-y_2y_1'\) is
The solution of \[ \frac{dy}{dx} + \left[2(x-1)\sin x+x(x-2)\cos x\right]y = \frac{e^{2x\sin x}}{e^{x^2\sin x}}, \] at \(x=2\) if \(y(\pi)=0\) is
The initial value problem \[ (x-x^2)\frac{dy}{dx} = (2x-1)y, \qquad y(x_0)=y_0, \] has a unique solution if \((x_0,y_0)\) equals to
The particular integral of the differential equation \[ (D^2+1)y = \sin x\,\sin2x \] is
The general solution of \[ \log\left(\frac{dy}{dx}\right)=ax+by \] is
If \(y_1,\;y_2\) are two dependent solutions of a second order linear homogeneous differential equation then \(y_1y_2'-y_2y_1'\) is
The solution of \[ \frac{dy}{dx} + \left[2(x-1)\sin x+x(x-2)\cos x\right]y = \frac{e^{2x\sin x}}{e^{x^2\sin x}}, \] at \(x=2\) if \(y(\pi)=0\) is
If the integrating factor of $\frac{dy}{dx} + \left[(x^2 - 2x)\cos x + 2(x - 1)\sin x\right]y = x^2$ is $e^{f(x)}$, then $f(3) =$}
If the particular integral of \( y'' - 4y' = x^2 e^{2x} \) is in the form \( y_p = e^{2x} y(x) \), then \( y(x) \) is
If the integrating factor of $\frac{dy}{dx} + \left[(x^2 - 2x)\cos x + 2(x - 1)\sin x\right]y = x^2$ is $e^{f(x)}$, then $f(3) =$}
If the particular integral of \( y'' - 4y' = x^2 e^{2x} \) is in the form \( y_p = e^{2x} y(x) \), then \( y(x) \) is
Given that \[ \frac{dy}{dx}=ye^x \] and when \[ x=0,\quad y=e, \] then the value of \(y\) (\(y>0\)) when \(x=1\) is
Solve the differential equation \( y e^y dx = (y^3 + 2x e^y) dy \), given that \( y(0) = 1 \).