- #1
asdf1
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for this O.D.E. :
y`= (x^2 + y^2)/xy
it's unseparable, so what other methods can there be taken?
y`= (x^2 + y^2)/xy
it's unseparable, so what other methods can there be taken?
Whoopsie, you've forgot a factor 1/2 on the left side.asdf1 said:here's my work:
y`=x/y +x/y
suppose y=vx
=>v=y/x
=> y`=v+xv`
so v+xv`=1/v+v
=> vdv=dx/x
=> v^2=ln[absolute value(x)]+c`
An ODE, or ordinary differential equation, is an equation that relates a function to its derivatives. It is used to model many real-world phenomena in science and engineering.
The equation "Y'= (x^2 + y^2)/xy" represents a first-order ODE, as it only involves the first derivative of the function y. It is a separable ODE, meaning that the variables can be separated and solved individually.
To solve this ODE, you can use the method of separation of variables. This involves isolating the variables on each side of the equation and then integrating both sides to find the general solution. From there, you can use initial conditions to find a particular solution.
This type of ODE can model various phenomena, such as population growth, chemical reactions, and electrical circuits. It is also commonly used in physics and engineering to describe the motion of objects.
Like any mathematical model, this ODE has limitations. It may not accurately represent all real-world situations, and the solutions may not always be physically meaningful. Additionally, some ODEs may be difficult or impossible to solve analytically, requiring numerical methods instead.