1
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
$$n\left( x \right) = {k^2}\,\exp \left( {{{ - x} \over {\sqrt L }}} \right)$$
2
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
$$P - 3,\,Q - 2,\,R - 1,S - 2$$
3
GATE ECE 2008
+2
-0.6
Which of the following is a solution to the differential equation $${d \over {dt}}x\left( t \right) + 3x\left( t \right) = 0,\,\,x\left( 0 \right) = 2?$$
A
$$x\left( t \right) = 3{e^{ - t}}$$
B
$$x\left( t \right) = 2\,{e^{ - 3t}}\,$$
C
$$x\left( t \right) = {{ - 3} \over 2}{t^2}$$
D
$$x\left( t \right) = 3{t^2}$$
4
GATE ECE 2007
+2
-0.6
The solution of the differential equation $${k^2}{{{d^2}y} \over {d\,{x^2}}} = y - {y_2}\,\,$$ under the boundary conditions (i) $$y = {y_1}$$ at $$x=0$$ and (ii) $$y = {y_2}$$ at $$x = \propto$$ where $$k$$, $${y_1}$$ and $${y_2}$$ are constant is
A
$$y = \left( {{y_1} - {y_2}} \right){e^{{{ - x} \over {{k^2}}}}} + {y_2}$$
B
$$y = \left( {{y_2} - {y_1}} \right){e^{{{ - x} \over k}}} + {y_1}$$
C
$$y = \left( {{y_1} - {y_2}} \right)\,\sin \,h\left( {{x \over k}} \right) + {y_1}$$
D
$$y = \left( {{y_1} - {y_2}} \right){e^{{{ - x} \over k}}} + {y_2}$$
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