I don't know shit at all about differential equations, but if you can get a funtion of x one one side of the equation and a function of y on the other side you can integrate and find a solution. Something like this
[ math ]
dy/dx=ky
dy/y=kdx
ln(y)=kx+c
y=e^(kx)e^c
y=e^(kx)C
[ /math ]
Where k is a constant, c is the constant of integration and C=e^c is another constant

>>16956060
I might be retarded

>>16956060
Wait let me try

[math]
dy/dx=ky
dy/y=kdx
ln(y)=kx+c
y=e^(kx)e^c
y=e^(kx)C
[/math]

>>16956080
that works for the homogeneous solution, bit it doesn't for the particular one.

>>16956029
integrating factors aren't hard, just get some practice

>>16956029
The actual reason for this is experimental. You observe the system you are attempting to model, collect data, and make out patterns. Then you model said patterns in some mathematical model, often as a function in some variables. Then you turn to the DE(s) that encapsulate current theory and input your model function as a trial solution to see if it's consistent with the DE. If yes, it's a solution to the DE and thus your experimental observation is a theoretical prediction. As an example, harmonic oscillators of many kinds follow a generally sinusoidal curve, so input such a curve as a trial solution to the wave equation.

Nowadays analytically solving DEs like this is not possible with complex physics theories, and numerically simulating the DEs of a theory is the norm thanks to computers. Then the process is simpler: is the data from experiment equivalent to data from the theoretical simulation within statistical uncertainty?

>>16956029
>Prove theorems on existence and uniqueness of solutions
>Reduces problem to finding some collection of solutions
>Can write any analytic function as a power series about a point
>Can account for regular singularities by multiplying by some factor
>etc.
The only interesting part of differential equations are the existence and uniqueness theorems. Everything else is messing around with equations, that only physicists enjoy.