Method of undetermined coefficients

In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, a 'guess' is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time consuming to perform. ∑ i = 0 n K i x i {displaystyle sum _{i=0}^{n}K_{i}x^{i}!} K cos ⁡ ( a x ) + M sin ⁡ ( a x ) {displaystyle Kcos(ax)+Msin(ax)!} e a x ( K cos ⁡ ( b x ) + M sin ⁡ ( b x ) ) {displaystyle e^{ax}(Kcos(bx)+Msin(bx))!} ( ∑ i = 0 n Q i x i ) cos ⁡ ( b x ) + ( ∑ i = 0 n R i x i ) sin ⁡ ( b x ) {displaystyle left(sum _{i=0}^{n}Q_{i}x^{i} ight)cos(bx)+left(sum _{i=0}^{n}R_{i}x^{i} ight)sin(bx)} e a x ( ( ∑ i = 0 n Q i x i ) cos ⁡ ( b x ) + ( ∑ i = 0 n R i x i ) sin ⁡ ( b x ) ) {displaystyle e^{ax}left(left(sum _{i=0}^{n}Q_{i}x^{i} ight)cos(bx)+left(sum _{i=0}^{n}R_{i}x^{i} ight)sin(bx) ight)} In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, a 'guess' is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time consuming to perform. Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.