Differential Equations · Engineering Mathematics · GATE ME
Marks 1
For the differential equation given below, which one of the following options is correct?
$$ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0,0 \leq x \leq 1,0 \leq y \leq 1 $$
Solution of ∇2T = 0 in a square domain (0 < x < 1 and 0 < y < 1) with boundary conditions:
T(x, 0) = x; T(0, y) = y; T(x, 1) = 1 + x; T(1, y) = 1 + y is
The value of $$y$$ at $$t=3$$ is
Marks 2
Let $y$ be the solution of the differential equation with the initial conditions given below. If $y(x=2)=A \ln 2$, then the value of $A$ is _________ (rounded off to 2 decimal places).
$$ x^2 \frac{d^2 y}{d x^2}+3 x \frac{d y}{d x}+y=0 \quad y(x=1)=0 \quad 3 x \frac{d y}{d x}(x=1)=1 $$
If $x(t)$ satisfies the differential equation
$t \frac{dx}{dt} + (t - x) = 0$
subject to the condition $x(1) = 0$, then the value of $x(2)$ is __________ (rounded off to 2 decimal places).
Consider the second-order linear ordinary differential equation
$\rm x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=0, x\ge1$
with the initial conditions
$\rm y(x=1)=6, \left.\frac{dy}{dx}\right|_{x=1}=2$
The value of 𝑦 at 𝑥 = 2 equals ________.
(Answer in integer)
For the exact differential equation,
$\frac{du}{dx}=\frac{-xu^2}{2+x^2u}$
which one of the following is the solution?
$$\,\,{{{d^2}x\left( t \right)} \over {d{t^2}}} + x\left( t \right) = 0,t > 0,\,\,$$ such that
$$\,{x_1}\left( 0 \right) = 1,{\left. {{{d{x_1}\left( t \right)} \over {dt}}} \right|_{t = 0}} = 0,$$ $$\,\,\,\,{x_2}\left( 0 \right) = 0,{\left. {{{d{x_2}\left( t \right)} \over {dt}}} \right|_{t = 0}} = 1$$
The wronskian $$\,w\left( t \right) = \left| {{{\matrix{ {{x_1}\left( t \right)} \cr {d{x_1}\left( t \right)} \cr } } \over {dt}}} \right.\left. {{{\matrix{ {{x_2}\left( t \right)} \cr {d{x_2}\left( t \right)} \cr } } \over {dt}}} \right|$$ at $$\,\,t = \pi /2$$