8/6/2019 130001-2 Basic Electronics gtu 3rd sem paper
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Q.3 (a) Solve the initial value problem :02 =+ y y y , 4)0( = y
and 5)0( = y
05
(b)Given the functions
xe and xe
on any interval [a, b].Are these functions linearly independent
or dependent?
04
(c) Using the method of variation of parameter solve the
differentialequation: x y y sec=+ .
05
ORQ.3 (a)Prove that: )()]([ 11
1 x J x x J xdxd
nn
nn +
++ = .
05
(b) Attempt ( any three ). 09(i) Express the polynomial 32 23 +
x x x in terms of Legendre
polynomials.
(ii) Show that
=1
1
0)()( dx xP xP nm , if nm .
(iii) By using generating relation of Legendre polynomials,
evaluate
)1(nP .
(iv) Obtain the value of
=1
1
2 0)( dx xPn .
Q.4 (a) Find the Fourier series of the function
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(c)(i) Evaluate: +
22
21
ses
Ls
.
(ii) Using convolution theorem, obtain the value of +
)4(1
21
ss L .
06
ORQ.5 (a) Find the solution ),( y xu of the partial differential
equation
0=+ yy xx vu by method of separation of variables.
07
(b) Attempt ( any one ).(i) Prove that Laplacian u in polar
coordinate is
2
2
22
22 11
+
+
= u
r xu
r xu
u .
(ii) Find the potential inside a spherical capacitor consisting
two metallichemispheres of radius 1 ft separated by a small slit
for reasons of insulation , if the upper hemisphere is kept at 110
volts and lowerhemisphere is grounded.
