Mathematics UPSC PYQ 2021

Find the general solution of the differential equation : x^2 \frac{d^2 y}{dx^2} - 2x(1+x)\frac{dy}{dx} + 2(1+x)y = 0. Hence, solve the differential equation : x^2 \frac{d^2 y}{dx^2} - 2x(1+x)\frac{dy}{dx} + 2(1+x)y = x^3 by the method of variation of parameters.

A heavy string, which is not of uniform density, is hung up from two points. Let T_1, T_2, T_3 be the tensions at the intermediate points A, B, C of the catenary respectively where its inclinations to the horizontal are in arithmetic progression with common difference \beta. Let \omega_1 and \omega_2 be the weights of the parts AB and BC of the string respectively. Prove that (i) Harmonic mean of T_1, T_2 and T_3 = \frac{3T_2}{1 + 2\cos \beta} (ii) \frac{T_1}{T_3} = \frac{\omega_1}{\omega_2}

Obtain the partial differential equation by eliminating arbitrary function f from the equation f(x + y + z, x^2 + y^2 + z^2) = 0.