(adsbygoogle = window.adsbygoogle || []).push({}); differential equations - 2nd order homogenous eq'n

sorry the title should read 2nd order homogenous eq'n, not nonhomogenous

Find the general solution of the equation:

(1+t^2)d^2y/dt^2 - 2t dy/dt + 2y = 0, given that y1(t) = t is one solution.

My attempt:

divided equation by 1+t^2

d^2y/dt^2 - 2t/1+t^2 dy/dt + 2/1+t^2 y = 0

using u(t) eq'n given:

u(t) = exp(-integ(- 2t/t+t^2 dt)) / y1^2(t)

let x = 1+t^2

dx/dt = 2t

dt = dx/2t

u(t) = exp(integ(2t/x dx/2t)) / t^2

u(t) = exp(integ(1/x dx)) / t^2

u(t) = exp(ln x) / t^2

u(t) = e^ln(1+t^2) / t^2

u(t) = e^ln(1+t^2) / t^2

u(t) = (1+t^2) / t^2

u(t) = (1+t^2)/(t^2)

using y2(t) eq'n given:

y2(t) = t integ (u(t)dt)

y2(t) = t integ ((1+t^2)/(t^2)dt)

y2(t) = t [ integ(1/t^2 dt) + integ (1 dt)]

y2(t) = t[-1/t + t]

y2(t) = -1 + t^2

y(t) = c1y1 + c2y2

y(t) = c1t + c2[-1 + t^2]

y(t) = c1t - c2[t^2 - 1]

can someone please confirm whether i am doing this correct?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Differential equations - 2nd order nonhomogenous eq'n

**Physics Forums | Science Articles, Homework Help, Discussion**