Page 43 - Schaum's Outline of Differential Equations
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26 SEPARABLE FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 4
This last equation is separable; its solution is
which, when evaluated, yields v = In \x\ - c, or
where we have set c = -In \k\ and have noted that In \x\ + In \ls\ = In Ifcd. Finally, substituting v = ylx back into (_/),
we obtain the solution to the given differential equation as y = x In Ifcd.
4.12. Solve
This differential equation is not separable. Instead it has the form y' =f(x, y), with
where
so it is homogeneous. Substituting Eqs. (4.6) and (4.7) into the differential equation as originally given, we obtain
which can be algebraically simplified to
This last equation is separable; its solution is
Integrating, we obtain in or
where we have set c = -In \k\ and then used the identities
Finally, substituting v = ylx back into (_/), we obtain
4.13. Solve the differential equation of Problem 4.12 by using Eqs. (4.9) and (4.10
We first rewrite the differential equation as
Then substituting (4.9) and (4.10) into this new differential equation, we obtain
