1
GATE ECE 2014 Set 4
Numerical
+2
-0
With initial values $$\,\,\,y\left( 0 \right) = y'\left( 0 \right) = 1,\,\,\,$$ the solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 4y = 0\,\,$$ at $$x=1$$ is ________.
2
GATE ECE 2014 Set 3
+2
-0.6
Which ONE of the following is a linear non - homogeneous differential equation , where $$x$$ and $$y$$ are the independent and dependent variables respectively?
A
$${{dy} \over {dx}} + xy = {e^{ - x}}$$
B
$${{dy} \over {dx}} + xy = 0$$
C
$${{dy} \over {dx}} + xy = {e^{ - y}}$$
D
$${{dy} \over {dx}} + {e^{ - y}} = 0$$
3
GATE ECE 2010
+2
-0.6
A function $$n(x)$$ satisfies the differential equation $${{{d^2}n\left( x \right)} \over {d{x^2}}} - {{n\left( x \right)} \over {{L^2}}} = 0$$ where $$L$$ is a constant. The boundary conditions are $$n(0)=k$$ and $$n\left( \propto \right) = 0.$$ The solution to this equation is
A
$$n\left( x \right) = k\,\exp \left( {{{ - x} \over L}} \right)$$
B
$$n\left( x \right) = k\,\exp \left( {{{ - x} \over {\sqrt L }}} \right)$$
C
$$n\left( x \right) = {k^2}\,\exp \left( {{{ - x} \over L}} \right)$$
D
$$n\left( x \right) = {k^2}\,\exp \left( {{{ - x} \over {\sqrt L }}} \right)$$
4
GATE ECE 2009
+2
-0.6
Match each differential equation in Group $$I$$ to its family of solution curves from Group $$II.$$

Group $$I$$
$$P:$$$$\,\,\,$$ $${{dy} \over {dx}} = {y \over x}$$
$$Q:$$$$\,\,\,$$ $${{dy} \over {dx}} = {{ - y} \over x}$$
$$R:$$$$\,\,\,$$ $${{dy} \over {dx}} = {x \over y}$$
$$S:$$$$\,\,\,$$ $${{dy} \over {dx}} = {{ - x} \over y}$$

Group $$II$$
$$(1)$$$$\,\,\,$$ Circle
$$(2)$$$$\,\,\,$$ straight lines
$$(3)$$$$\,\,\,$$ Hyperbola

A
$$P - 2,\,Q - 3,\,R - 3,S - 1$$
B
$$P - 1,\,Q - 3,\,R - 2,S - 1$$
C
$$P - 2,\,Q - 1,\,R - 3,S - 3$$
D
$$P - 3,\,Q - 2,\,R - 1,S - 2$$
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