1
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
$$y = \left( {{y_1} - {y_2}} \right){e^{{{ - x} \over k}}} + {y_2}$$
2
GATE ECE 2006
+2
-0.6
For the differential equation $${{{d^2}y} \over {d{x^2}}} + {k^2}y = 0,$$ the boundary conditions are
(i) $$y=0$$ for $$x=0$$ and
(ii) $$y=0$$ for $$x=a$$
The form of non-zero solution of $$y$$ (where $$m$$ varies over all integrals ) are
A
$$y = \sum\limits_m {{A_m}\sin \left( {{{m\pi x} \over a}} \right)}$$
B
$$y = \sum\limits_m {{A_m}\cos \left( {{{m\pi x} \over a}} \right)}$$
C
$$y = \sum\limits_m {{A_m}\,\,{X^{{{m\pi x} \over a}}}}$$
D
$$y = \sum\limits_m {{A_m}\,\,{e^{{{m\pi x} \over a}}}}$$
3
GATE ECE 2001
Subjective
+2
-0
Solve the differential equation $${{{d^2}y} \over {d{x^2}}} + y = x\,\,$$ with the following conditions $$(i)$$ at $$x=0, y=1$$ $$(ii)$$ at $$x=0,$$ $${y^1} = 1$$
4
GATE ECE 1994
+2
-0.6
Match each of the items $$A, B, C$$ with an appropriate item from $$1, 2, 3, 4$$ and $$5$$

List-$${\rm I}$$
$$(P)$$ $${a_1}{{{d^2}y} \over {d{x^2}}} + {a_2}y{{dy} \over {dx}} + {a_3}y = {a_4}$$
$$(Q)$$ $${a_1}{{{d^2}y} \over {d{x^3}}} + {a_2}y = {a_3}$$
$$(Q)$$ $${a_1}{{{d^2}y} \over {d{x^2}}} + {a_2}x{{dy} \over {dx}} + {a_3}{x^2}y = 0$$

List-$${\rm II}$$
$$(1)$$ Non-linear differential equation
$$(2)$$ Linear differential equation with constants coefficients
$$(3)$$ Linear homogeneous differential equation
$$(4)$$ Non-linear homogeneous differential equation
$$(5)$$ Non-linear first order differential equation

A
$$P - 1,Q - 2,R - 3$$
B
$$P - 3,Q - 4,R - 2$$
C
$$P - 2,Q - 5,R - 3$$
D
$$P - 3,Q - 1,R - 2$$
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