general solution of elastic beam equation For the following equation;

$t^4 \dfrac{d^2u}{dt^2} + \lambda^2 u = 0, \quad \lambda >0, ~ t>0,$

where $u(t)$ is real valued function. by using the change of variables 

$t=\dfrac{1}{\tau} , \quad u(t)=\dfrac{v(\tau)}{\tau} ,$

how can we find the general solution of the above equation.

This shows a method to carry out the change of variable and the change of function, as requested in the wording. Then, you can solve the ODE which is obtained on a well-known form :

