Using the Bernoulli method solve the differential equation t

Using the Bernoulli method, solve the differential equation: t^2 dy/dt + y^2 = ty.

Solution

t^2y\'+y^2=ty

t^2y\'-ty=-y^2

n=2 is highest power of y

So substitution is

u=y^{1-n}=y^{1-2}=1/y

y=1/u

y\'=-u\'/u^2

SUbstituting gives

t^2y\'-ty=-y^2

-t^2u\'/u^2-t/u=-1/u^2

t^2u\'+tu=1

tu\'+u=1/t

(tu)\'=1/t

Integrating gives

tu=ln(t)+C

u=(ln(t)+C)/t

Hence,

y=t/(C+ln(t))