twice gives Finally, dividing by the integrating factor gives Solving nth order linear differential equations Integrating factors can be extended to any order, though the form of the equation needed to apply them gets more and more specific as order increases, making them less useful for orders 3 and above. The general idea is to differentiate the function times for an th order differential equation and combine like terms. This will yield an equation in the form If an th order equation matches the form that is gotten after differentiating times, one can multiply all terms by the integrating factor and integrate times, dividing by the integrating factor on both sides to achieve the final result. Example A third order usage of integrating factors gives thus requiring our equation to be in the form For example in the differential equation we have , so our integrating factor is . Rearranging gives Integrating thrice and dividing by the integrating factor yields See also Variation of parameters Differential equations Product rule Quotient rule Exact differential Matrix exponential External links . Ordinary differential equations de:Exakte Differentialgleichung