# How to know if there's an explicit solution for differential equation

#### find_the_fun

##### Active member
Solve the given differential equation by separation of variables

$$\displaystyle y\ln{x}\frac{dx}{dy}=(\frac{y+1}{x})^2$$

I got it down to

$$\displaystyle \ln{x}x^3-\frac{x^3}{3}=y^3+3y+y^2$$

At this point I had no idea how to solve having y^3 y^2 and y terms so I did what any good student would do and checked the back of the book. The answer given was basically the same as I had got. My question is how do you know when you are done?

#### MarkFL

Staff member
I get a different result:

$$\displaystyle \frac{y^2}{2}+2y+\ln|y|=\frac{x^3}{9}\left(3\ln(x)-1 \right)+C$$

I would look at the fact that we can solve for neither variable in terms of elementary functions because both are both inside and outside of log functions. So, I would leave the solution implicitly defined.

#### Prove It

##### Well-known member
MHB Math Helper
Solve the given differential equation by separation of variables

$$\displaystyle y\ln{x}\frac{dx}{dy}=(\frac{y+1}{x})^2$$

I got it down to

$$\displaystyle \ln{x}x^3-\frac{x^3}{3}=y^3+3y+y^2$$

At this point I had no idea how to solve having y^3 y^2 and y terms so I did what any good student would do and checked the back of the book. The answer given was basically the same as I had got. My question is how do you know when you are done?
It's a separable equation...

\displaystyle \begin{align*} y\ln{(x)}\,\frac{dx}{dy} &= \left( \frac{y + 1}{x} \right) ^2 \\ y\ln{(x)}\,\frac{dx}{dy} &= \frac{y^2 + 2y + 1}{x^2} \\ x^2\ln{(x)}\,\frac{dx}{dy} &= \frac{y^2 + 2y + 1}{y} \\ x^2\ln{(x)}\,\frac{dx}{dy} &= y + 2 + \frac{1}{y} \end{align*}

Now you can integrate both sides with respect to y.