wmccunes said:I haven't done ODEs in a while nor have a book handing.
How do I tackle an equation of the form
\[
2xyy'=-x^2-y^2
\]
I tried polar but that didn't seem to work.
Separable ODEs can be expressed as a product of two functions, one of which depends only on the independent variable and the other only on the dependent variable. Non-separable ODEs cannot be expressed in this way and may require more complex methods to solve.
To solve a separable ODE, you can use the method of separation of variables. This involves isolating the dependent variable on one side of the equation and the independent variable on the other side, then integrating both sides to find the general solution.
Yes, in some cases it is possible to manipulate a non-separable ODE into a separable form by using techniques such as substitution or integration by parts. However, this may not always be possible and other methods of solving the ODE may need to be used.
Separable ODEs are commonly used in physics and engineering to model physical systems such as population growth, radioactive decay, and projectile motion. Non-separable ODEs can be found in more complex systems, such as fluid dynamics, chemical reactions, and electrical circuits.
An ODE is separable if it can be expressed in the form of dy/dx = g(x) * h(y), where g(x) and h(y) are functions of x and y respectively. If an ODE cannot be written in this form, it is non-separable.