Question

Consider the differential equation
`x^(2) (dy)/(dx) =6 y^(2) +6xy` which may be considered either as a homogenous equation or as a Bernoulli equation.

If we make the substitution
`y(x)=xv(x)` relevant to homogenous equations, we obtain
`(dv)/(dx)=`
If we make the substitution
`z(x)= (y(x))^(-1)` relevant to homogenous equations, we obtain
`(dz)/(dx)=`
Using either (or both) of these methods, solve the initial value problem for the above equation where y(3)=6. Find the interval of validity of this solution.

Answer 1

As a Bernoulli equation:


`x^2(\mathrm dy)/(\mathrm dx)=6y^2+6xy\iff x^2y^(-2)(\mathrm dy)/(\mathrm dx)-6xy^(-1)=6`

Let
`z=y^(-1)\implies(\mathrm dz)/(\mathrm dx)=-y^(-2)(\mathrm dy)/(\mathrm dx)`. The ODE becomes


`-x^2(\mathrm dz)/(\mathrm dx)-6xz=6`

`x^6(\mathrm dz)/(\mathrm dx)+6x^5z=-6x^4`

`(\mathrm d)/(\mathrm dx)[x^6z]=-6x^4`

`x^6z=-6\displaystyle\int x^4\,\mathrm dx`

`x^6z=-\frac65x^5+C`

`z=-\frac6{5x}+\frac C{x^6}`

`y^(-1)=-\frac6{5x}+\frac C{x^6}`

`y=\frac1{\frac C{x^6}-\frac6{5x}}`

`y=(5x^6)/(C-6x^5)`

With
`y(3)=6`, we get


`6=(5(3)^6)/(C-6(3)^5)\implies C=\frac{4131}2`

so the solution is


`y=\frac{5x^6}{\frac{4131}2-6x^5}=(10x^6)/(4131-12x^5)`

which is valid as long as the denominator is not zero, which is the case for all
`x\\eq\sqrt[5]{(4131)/(12)}`.