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<e> A string of iength L is stretched and fastened to two fix points. Find the solution of
the r.{ave equatiorl (vibrating string) ytt = a^2.yxx, when initial displacernent
y(x,0) = f (x) = b sin (pi.x / t).
also find the Fourier cosine transformation of exp(-x^2)
<e> Solve the first order linear inhomogeneous differential equation using the constant variation method
y,- (3y/x)=x
<e> a) Establish the relation ๐
๐
๐๐ฅ
๐
(๐ฅ
2 โ 1)
๐ = ๐! 2
๐ ๐๐(๐ฅ)
b) Use Laplace transform to solve
๐
2๐ฆ
๐๐ก
2 + 2
๐๐ฆ
๐๐ก + ๐ฆ = ๐ก๐
โ๐ก ๐ค๐๐กโ ๐ฆ(0) = 1, ๐ฆ
โฒ
(0) = โ2
<e>Find a power series solution of ๐ฅ๐ฆโฒ=๐ฆ .
(a) The differential equation (2๐ฅ 2 + ๐๐ฆ 2 )๐๐ฅ + ๐๐ฅ๐ฆ ๐๐ฆ = 0 can made exact by multiplying with integrating factor 1 ๐ฅ โ 2.
Then find the relation between ๐ and ๐. (b) Find one fourth roots of unity
verify that -2x^2y+y^2=1 is the implicit solution of the of the differential equation (x^2-y)dy/dx + 2xy=0
determine the general/particular solution for each equation using the applicable solution to equations of order one (separable, homogenous,linear,exact)
x^2 dy/dx + sinx - y = 0
determine the general/particular solution for each equation using the applicable solution to equations of order one (separable, homogenous,linear,exact)
(x^2-xy+y^2)dx-xy dy = 0
determine the general/particular solution for each equation using the applicable solution to equations of order one (separable, homogenous,linear,exact)
3x(xy-2)dx + (x^3=2y)dy = 0
determine the general/particular solution for each equation using the applicable solution to equations of order one (separable, homogenous,linear,exact)
- (1+y^2)dx + (1=x^2)dy = 0 ; when x = 0, y = 1
