Inseparable differential equation
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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.[citation needed]
Examples[]
Consider the general inseparable equation
Now an integrating factor μ is defined as[1][note 1]
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, i.e., linear differential equations. 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 and Then,
It follows that
or
Now one can just take the inverse Laplace transform of Y(s) to get the solution y to the original equation.
See also[]
- Differential equation
- Examples of differential equations
- Numerical methods for ordinary differential equations
- Method of variation of parameters
Notes[]
- ^ The integrating factor may also be called ρ or r in different textbooks.
References[]
- ^ Edwards, C. Henry (2008). Elementary differential equations with boundary value problems. David E. Penney, David Calvis (Sixth ed.). Upper Saddle River, NJ: Pearson Education. pp. 46–48. ISBN 978-0-13-600613-8. OCLC 234257278.
- Ordinary differential equations