Example 2 Solvethe differentialequation.
Solution. We have and a substitution gives
.
Exercise
- Solvethe differentialequation with.
- Solvethe differentialequation.
Hint: Let.
SolvingSeparableFirst-OrderDifferentialEquations
The next type of differentialequationwhereanalyticsolutionare rela- tivelyeasyis whenthe dependence
of on and are separable: where is the productof a
functionsof and respectively. The solutionis in the form. Here is never or
the valuesof in the solutionswill be restrictedby where.
Example 1 Solvethe differentialequationy'=xywith the initialconditiony(0) = 1.
Solution.Separatingxandyturnsthe equationin differentialform. Integratingboth sides,we
have.
Theny(0) = 1 gives , i.e.C= 0 and.
So
Therefore,the solutionsare.
HereQ(y) =yis 0 wheny= 0 and the valuesofyin the solutionssatisfyy> 0 ory< 0.
Example2.Solvethe differentialequation 2 xy'= 1 - y^2.
Solution.Separatingxandyturnsthe equationin differentialform