Answer to Question #50907 in Differential Equations for Nayan Jyoti Das

Question

8. a) Solve the differential equation x^3 p^2 + x^2 yp + a^3 = 0 and also obtain its singular solution, if
it exists.

b) The differential equation satisfied by a beam uniformly loaded (W kg /meter) , with one
end fixed and the second end subjected to a tensile force-P, is given by
EI (d^2 y / d x^2) = Py - 1/2 W x^2 ,

where E is the modulus of elasticity and I is the moment of inertia. Show that the elastic
curve for the beam with conditions y = 0 and dy/dx = 0 at x = 0 , is given by

y = W / Pn^2 (1-cosh nx) + W x^2 / 2P , where n^2 = (P/EI)

c) For 0 < x < 5 and t > 0 , solve the one-dimensional heat flow equation d u/d t = 4 d^2 u/ d x^2
satisfying the conditions u(t, 0) = u(t,5) = 0, u(0, x) = x