Differential Equations – Separating Variables

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It is a very unusual thing to be given a differential equation that will just, well, integrate. Usually some manipulations must be performed, whether it is simplifying, grouping like terms, simplifying, making substitutions or separating variables – the technique illustrated here. In general a differential equation may haveandterms on both sides, but if the equation is of a certain form –– we can rearrange to have all terms includingon the right hand side and all terms includingon the left hand side, obtaining in this case,

We can then integrate both sides:

Example: Solve the differential equation

Multiply byand divide byto giveWe can now integrate:

is a product which we integrate by parts obtainingTo find the constantwe need what is called a boundary condition. Suppose then that we have that whenSubstitute these values into (1) to obtain

hence

Example: Solve

Factorise the right hand side intoto give which is separable.

Multiply byand divide byto giveWe can now integrate:(1)

If we are to makethe subject we exponentiate both sides, raisingto the power of both sides:

whereNotice how the constant termin (1) becomes the constant factorwhen we exponentiate.

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