Find the function y=y(x), for x>0 which satisfies the differential equation x (dy)/(dx) -2y= x^(12) , (x\ \textgreater \ 0) with the initial condition y(1)=10

Question

Find the function y=y(x), for x>0 which satisfies the differential equation


`x (dy)/(dx) -2y= x^(12) , (x\ \textgreater \ 0)` with the initial condition y(1)=10

Answer 1

`x(\mathrm dy)/(\mathrm dx)-2y=x^(12)`

`\frac1{x^2}(\mathrm dy)/(\mathrm dx)-\frac2{x^3}y=x^9`

`(\mathrm d)/(\mathrm dx)\left[\frac1{x^2}y\right]=x^9`

`\frac1{x^2}y=\displaystyle\int x^9\,\mathrm dx`

`\frac1{x^2}y=\frac1{10}x^(10)+C`

`y=\frac1{10}x^(12)+Cx^(-2)`

With
`y(1)=10`, we have


`10=\frac1{10}1^(12)+C(1)^(-2)\implies C=(99)/(10)`

and so the particular solution is


`y=\frac1{10}x^(12)+(99)/(10x^2)`