# Toronto Math Forum

## MAT244--2020F => MAT244--Lectures & Home Assignments => Chapter 2 => Topic started by: Julian on September 28, 2020, 12:57:40 PM

Title: W2L3 What's u in the general solution to a homogeneous equation?
Post by: Julian on September 28, 2020, 12:57:40 PM
In tutorial with Neall last week, we were asked to find the general solution to the equation $(xy +y^2+x^2)dy-x^2dx=0$. Based on techniques from class (letting $y=ux$), I determined that the parametric general solution to this (homogeneous) equation is $x=Ce^{\arctan(u)}$ and $y=Cxe^{\arctan(u)}$ or equivalently $x=Ce^{\arctan(\tfrac{y}{x})}$ and $y=Cxe^{\arctan(\tfrac{y}{x})}$.

My question is why is this satisfactory? Shouldn't we find a function $y$ purely in terms of $x$ instead? Indeed, in tutorial, we were suggested that $y=\tan(\ln(Cx))x$. It's really not obvious to me that these are even equivalent. (If anyone can tell me why these are equivalent, I would also appreciate that.)
Title: Re: W2L3 What's u in the general solution to a homogeneous equation?
Post by: Victor Ivrii on September 29, 2020, 04:32:46 AM
It was explained in the lectures that solutions in the parametric form are also admissible. You could ask it on the tutorial.