The discussion centers on solving the separable differential equation (x + 2y) dy/dx = 1 with the initial condition y(0) = 1. Participants clarify that the equation cannot be separated and is not a standard ordinary differential equation (ODE). The correct approach involves rewriting the equation in the form dy/dx = 1/(x + 2y) and recognizing that it may require a substitution method, specifically y = ux, to explore its homogeneity. The conversation emphasizes the need for the equation to be expressed in a specific format before applying substitutions.
PREREQUISITESStudents and educators in mathematics, particularly those focusing on differential equations, as well as anyone seeking to deepen their understanding of solving complex ODEs.
ch2kb0x said:Problem is, I can't separate it. This might be a homogenous type? If so, how would I make it into the g(y/x) form.
Thank you.
You can write the equation as dy/dx = 1/(x + 2y), where the right side is a function of x and y. I'm just offering a suggested approach based on your first post. It may or may not work.ch2kb0x said:Okay, so since you said y = ux, I am thinking that this is a homogenous equation...
However, if it is a homogeneous equation, before we can plug in y = ux, aren't we suppose to first have the equation in the form of dy/dx = f(x,y), where there exists a function such that f(x,y) is expressed g(y/x).
Then, AFTEr we can do the y=ux thing. correct me if I am wrong.
Mark44 said:You can write the equation as dy/dx = 1/(x + 2y), where the right side is a function of x and y. I'm just offering a suggested approach based on your first post. It may or may not work.