# Solve the differential equation: y′′y′+yy′+yy′′=0

In summary, the conversation discusses the substitution of ##y=e^{\int z(x)}## in a first order differential equation, which leads to a complicated implicit solution. The participants also mention simple solutions and the possibility of finding an expression for all solutions, but they doubt its existence. They suggest using the inverse function of the implicit expression as the main part of the solution. The conversation ends with a question about integrating the inverse function without knowing it.
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.

Are you sure you haven't forgotten to mention something? WolframAlpha gives a monster as solution, so I checked some simple solutions. Of course we have all constants ##y(x)\equiv c##, and ##y(x)=ae^{bx}## yields ##b=-\frac{1}{2}\pm i\frac{\sqrt{3}}{2}##. And these are the easy ones where all three terms are more or less equal. Using the symmetries of the equation gave me something like ##1=(1+u(x))(1+v(x))## with ##u(x)=\frac{y}{y''}\, , \,v(x)=\frac{y'}{y''}##. So without additional assumptions on ##y''## this seems quite complicated. This might be most obvious in the notation ##u+v+uv=0##, which equals a sum and a product, two very different objects.

fresh_42 said:
Are you sure you haven't forgotten to mention something?
Sorry,please tell me if there is anything that I should mention but I forgot.

So,do you mean that this equation cannot be solved analytically?

By the way,I tried this substitution from this :

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fresh_42 said:
It can with simple solutions as those I mentioned. I doubt there is an expression for all solutions.
Have a look: https://www.wolframalpha.com/input/?i=y"y'+yy'+yy"=0
Well,I also doubt that there exist any methods to produce such an expression as shown.
According to Wolfram Alpha ,I think actually the main part of the solution consist of the inverse function of the implicit expression above.Maybe my thought still has a little bit of use...

Well,I also doubt that there exist any methods to produce such an expression as shown.
According to Wolfram Alpha ,I think actually the main part of the solution consist of the inverse function of the implicit expression above.Maybe my thought still has a little bit of use...
Or can I integrate the inverse function,even I don't know it?

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It represents the relationship between a function and its rate of change.

## 2. What is the process for solving a differential equation?

The process for solving a differential equation involves finding the general solution, which is a solution that includes all possible solutions, and then applying initial or boundary conditions to find a particular solution.

## 3. What is the order of a differential equation?

The order of a differential equation is the highest derivative present in the equation. For example, in the equation y′′y′+yy′+yy′′=0, the order is 2.

## 4. What is a second-order differential equation?

A second-order differential equation is a differential equation that contains a second derivative of the dependent variable. It can be written in the form y′′+p(x)y′+q(x)y=r(x), where p(x) and q(x) are functions of x and r(x) is a function of x or a constant.

## 5. What is the importance of solving differential equations?

Solving differential equations is important in many fields of science and engineering, as they can be used to model and understand various processes and phenomena. They also have practical applications in fields such as physics, biology, and economics.

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