estro said:I'm trying to solve the following ODE: [tex] ydx+(\frac {e^x}{y}-1)dy=0[/tex]
I tried to transfer this ODE into exact form but no luck.
Will appreciate any help.
estro said:I tried it before:
[tex]X=y, Y=e^x/y-1[/tex]
[tex]u'(x)+(\frac {X_y-Y_x}{Y})u(x)=0[/tex]
LCKurtz said:Are you sure about the sign on that numerator?
estro said:Bellow is formula that I found in my book:
[tex]u'(x)+(\frac {X_y-Y_x}{Y})u(x)=0[/tex]=
[tex]u'(x)-u(x)=0[/tex] => u=e^x
Can't see where is the problem.
A non-linear first order ODE (ordinary differential equation) is an equation that involves the first derivative of a function with respect to the independent variable, and the function itself is raised to a power or multiplied by another function. This makes it a non-linear equation, as the dependent variable is not directly proportional to the independent variable.
In a linear first order ODE, the dependent variable is directly proportional to the independent variable, and the equation can be solved using separation of variables. However, in a non-linear first order ODE, the dependent variable is not directly proportional to the independent variable, and different methods such as substitution or numerical approximation may be needed to solve it.
Non-linear first order ODEs are commonly used in physics, engineering, and other fields to model complex systems such as population growth, chemical reactions, and electrical circuits. They can also be used to study the behavior of biological systems, economics, and climate change.
There are several methods for solving non-linear first order ODEs, such as substitution, numerical approximation, and power series solutions. The method used will depend on the specific equation and the techniques known to the mathematician. In some cases, it may not be possible to find an exact solution and numerical methods may be used instead.
One of the main challenges in solving non-linear first order ODEs is that there is no general method that can be applied to all equations. Each equation may require a different approach, and some equations may not have an exact solution at all. Additionally, non-linear equations can exhibit chaotic behavior, making it difficult to predict the behavior of the system over time.