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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)
Find the integral surface of the linear partial differential equation x(x^2+z)p - y(y^2+z)q = (x^2-y^2)z; p, q has their usual meaning , which contains the straight line
Solve (𝐷
2 − 3𝐷 + 2)𝑦 = 𝑥
2 + sin 𝑥 where 𝐷 =
𝑑
𝑑𝑥
a) A thin metal plate bounded by the 𝑥-axis and the lines 𝑥 = 0 and 𝑥 = 1 and
stretching to infinity in the 𝑦-direction has its upper and lower faces perfectly
insulated and its vertical edges and edge at infinity are maintained at the constant
temperature 0 °𝐶, while over the base temperature of 50 °𝐶 is maintained. Find
the steady state temperature 𝑢(𝑥,𝑡).
[5]
b) If 𝑓(𝑥𝑦^2, 𝑧 − 2𝑥) = 0 then prove that 𝑥 𝑧x −1/2𝑦 𝑧𝑦 = 2 𝑥
Solve (𝐷^2 − 3𝐷 + 2)𝑦 = 𝑥^2 + sin 𝑥 where 𝐷 =𝑑/𝑑𝑥
Given x'=x(-20-x+2y)
y’=y(-50+x-y) identify the type of interaction represented by the system
solve the following differential equation: 2x^2y(d^2y/dx^2)+4y^2=x^2(dy/dx)^2+2xy(dy/dx)
Find orthogonal trajectory to the curve given by 𝑟 = 𝑎(1 + cos 𝜃)
Solve (𝐷
2 − 3𝐷 + 2)𝑦 = 𝑥
2 + sin 𝑥
a) A rod of length 𝐿 has its ends 𝐴 and 𝐵 maintained at 20 °𝐶 and 40 °𝐶 respectively
until steady state condition prevail. The temperature at 𝐴 is suddenly raised to
50 °𝐶 while that at 𝐵 is lowered to 10 °𝐶 and maintained thereafter. Find the
subsequent temperature distribution of the rod.
b) Solve (𝑥
2 − 𝑦
2 − 𝑧
2
)𝑝 + (2𝑥𝑦) 𝑞 = 2𝑥𝑧 where 𝑝 =
𝜕𝑧
𝜕𝑥 , 𝑞 =
𝜕𝑧
𝜕𝑦
