In mathematics, an inseparable differential equation is an ordinary differential equation that cannot be solved by using separation of variables. To solve an inseparable differential equation one can employ a number of other methods, like the Laplace transform, substitution, etc.
Examples
Consider the general inseparable equation
Now we will define a special factorial, μ as
Thus:
From here we can solve the equation using the above definition:
(using the product rule in reverse)
Finally, we obtain:
This can be used to solve most all inseparable equations containing no y to a degree other than one. For example, solving the inseparable equation:
By arranging in the form required, we obtain:
Now all that is necessary is to find the value of μ to plug into our original equation of
Plugging this into the original equation and simplifying gives us our final answer:
Consider for example the inseparable equation
Let us solve it using the Laplace transform. One has that
Using the convenience that Laplace transforms follow the rules of linearity, one can solve the above example for y by performing a Laplace transform on both sides of the differential equation, substituting in the initial values, solving for the transformed function, and then performing an inverse transform.
For the above example, assume initial values are
It follows that
or
Now one can just take the inverse Laplace transform of Y to get the solution y to the original equation.