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 (π·
2 β 3π· + 2)π¦ = π₯
2 + sin π₯ where π· =
π
ππ₯
a) A thin metal plate bounded by the π₯-axis and the lines π₯ = 0 and π₯ = 1 and
stretching to infinity in the π¦-direction has its upper and lower faces perfectly
insulated and its vertical edges and edge at infinity are maintained at the constant
temperature 0 Β°πΆ, while over the base temperature of 50 Β°πΆ is maintained. Find
the steady state temperature π’(π₯,π‘).
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b) If π(π₯π¦^2, π§ β 2π₯) = 0 then prove that π₯ π§x β1/2π¦ π§π¦ = 2 π₯
Solve (π·^2 β 3π· + 2)π¦ = π₯^2 + sin π₯ where π· =π/ππ₯
Given x'=x(-20-x+2y)
yβ=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 π = π(1 + cos π)
Solve (π·
2 β 3π· + 2)π¦ = π₯
2 + sin π₯
a) A rod of length πΏ has its ends π΄ and π΅ maintained at 20 Β°πΆ and 40 Β°πΆ respectively
until steady state condition prevail. The temperature at π΄ is suddenly raised to
50 Β°πΆ while that at π΅ is lowered to 10 Β°πΆ and maintained thereafter. Find the
subsequent temperature distribution of the rod.
b) Solve (π₯
2 β π¦
2 β π§
2
)π + (2π₯π¦) π = 2π₯π§ where π =
ππ§
ππ₯ , π =
ππ§
ππ¦