- #1

Baal Hadad

- 17

- 2

- Homework Statement
- Solve the following equation:

- Relevant Equations
- ##y''y'+yy'+yy''=0##

I tried the substitution ##y=e^{\int z(x)}##,##z(x)## is an arbitrary function to be determined.

Substitute this to the original differential equation,and dividing ##y^2## yields ##(z+1)z'+z^3+z^2+z=0##,which is a first order differential equation.

Trying to solve this first order differential equation yields ##x= ln z -\frac { ln (z^2+z+1)}{2} +\frac {\arctan( {\frac {2z+1}{\sqrt{3}}})}{{\sqrt{3}}}##

Then I can't continue here.The expression seems implicit and I cannot get an expression of ##z## in terms of ##x##,that can be substituted back to ##y=e^{\int z(x)}##.Had I use the wrong substitution or any better subsitutions?

Thanks.

Substitute this to the original differential equation,and dividing ##y^2## yields ##(z+1)z'+z^3+z^2+z=0##,which is a first order differential equation.

Trying to solve this first order differential equation yields ##x= ln z -\frac { ln (z^2+z+1)}{2} +\frac {\arctan( {\frac {2z+1}{\sqrt{3}}})}{{\sqrt{3}}}##

Then I can't continue here.The expression seems implicit and I cannot get an expression of ##z## in terms of ##x##,that can be substituted back to ##y=e^{\int z(x)}##.Had I use the wrong substitution or any better subsitutions?

Thanks.