Differential Equations
Practice QuestionsMarks 1
1
In the following differential equation, the numerically obtained value of $$y(t)$$, at $$t=1$$ is ___________ (Round off to 2 decimal places).
$${{dy} \over {dt}} = {{{e^{ - \alpha t}}} \over {2 + \alpha t}},\alpha = 0.01$$ and $$y(0) = 0$$
GATE EE 2023
2
With initial condition $$x\left( 1 \right)\,\,\, = \,\,\,\,0.5,\,\,\,$$ the solution of the differential equation, $$\,\,\,t{{dx} \over {dt}} + x = t\,\,\,$$ is
GATE EE 2012
3
With $$K$$ as constant, the possible solution for the first order differential equation $${{dy} \over {dx}} = {e^{ - 3x}}$$ is
GATE EE 2011
4
The solution of the first order differential equation $$\mathop x\limits^ \bullet \left( t \right) = - 3\,x\left( t \right),\,x\left( 0 \right) = {x_0}\,\,\,\,$$ is
GATE EE 2005
5
If at every point of a certain curve , the slope of the tangent equals $${{ - 2x} \over y},$$ the curve is _______.
GATE EE 1995
Marks 2
1
Consider ordinary differential equations given by $\dot{x}_1(t)=2 x_2(t), \dot{x}_2(t)=r(t)$ with initial conditions $x_1(0)=1$ and $x_2(0)=0$. If $r(t)=\left\{\begin{array}{ll}1, & t \geq 0 \\ 0, & t<0\end{array}\right.$, then $t=1, x_1(t)=$ _____________ (Round off to the nearest integer).
GATE EE 2025
2
Which of the following differential equations is/are nonlinear?
GATE EE 2024
3
Consider the differential equation $$\left( {{t^2} - 81} \right){{dy} \over {dt}} + 5ty = \sin \left( t \right)\,\,$$ with $$y\left( 1 \right) = 2\pi .$$ There exists a unique solution for this differential equation when $$t$$ belongs to the interval
GATE EE 2017 Set 1
4
Let $$y(x)$$ be the solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 4y = 0\,\,$$ with initial conditions $$y(0)=0$$ and $$\,\,{\left. {{{dy} \over {dx}}} \right|_{x = 0}} = 1.\,\,$$ Then the value of $$y(1)$$ is __________.
GATE EE 2016 Set 2
5
A function $$y(t),$$ such that $$y(0)=1$$ and $$\,y\left( 1 \right) = 3{e^{ - 1}},\,\,$$ is a solution of the differential equation $$\,\,{{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + y = 0\,\,$$ Then $$y(2)$$ is
GATE EE 2016 Set 1
6
A solution of the ordinary differential equation $$\,\,{{{d^2}y} \over {d{t^2}}} + 5{{dy} \over {dt}} + 6y = 0\,\,$$ is such that $$y(0)=2$$ and $$y(1)=$$ $$ - \left( {{{1 - 3e} \over {{e^3}}}} \right).$$ The value of $${{dy} \over {dt}}\left( 0 \right)$$ is
GATE EE 2015 Set 1
7
A differential equation $$\,\,{{di} \over {dt}} - 0.21 = 0\,\,$$ is applicable over $$\,\, - 10 < t < 10.\,\,$$ If $$i(4)=10,$$ then $$i(-5)$$ is
GATE EE 2015 Set 2
8
Consider the differential equation $${x^2}{{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}} - y = 0.\,\,$$ Which of the following is a solution to this differential equation for $$x > 0?$$
GATE EE 2014 Set 2
9
The solution for the differential equation $$\,\,{{{d^2}x} \over {d{t^2}}} = - 9x,\,\,$$ with initial conditions $$x(0)=1$$ and $${{{\left. {\,\,\,\,{{dx} \over {dt}}} \right|}_{t = 0}} = 1,\,\,}$$ is
GATE EE 2014 Set 1
10
For the differential equation $${{{d^2}x} \over {d{t^2}}} + 6{{dx} \over {dt}} + 8x = 0$$ with initial conditions $$x(0)=1$$ and $${\left( {{{dx} \over {dt}}} \right)_{t = 0}}$$ $$=0$$ the solution
GATE EE 2010
11
For the equation $$\,\,\mathop x\limits^{ \bullet \bullet } \left( t \right) + 3\mathop x\limits^ \bullet \left( t \right) + 2x\left( t \right) = 5,\,\,\,$$ the solution $$x(t)$$ approaches the following values as $$t \to \infty $$
GATE EE 2005