Higher-order linear differential equations often require special techniques to solve. The Method of Undetermined Coefficients and Variation of Parameters are two powerful approaches for finding particular solutions to nonhomogeneous equations .
These methods complement each other, with Undetermined Coefficients being simpler for specific forms of nonhomogeneous terms, while Variation of Parameters offers a more general approach. Understanding both expands your toolkit for tackling a wide range of differential equations.
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Top images from around the web for Polynomial and Exponential Forms Use the degree and leading coefficient to describe end behavior of polynomial functions ... View original
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Identify appropriate particular solution form based on nonhomogeneous term in linear differential equations
For polynomial nonhomogeneous terms use polynomial of same or higher degree (depending on characteristic equation roots)
Example: For x 2 + 3 x x^2 + 3x x 2 + 3 x , try A x 2 + B x + C Ax^2 + Bx + C A x 2 + B x + C
With exponential nonhomogeneous term e a x e^{ax} e a x , use form A e a x Ae^{ax} A e a x
Example: For 5 e 2 x 5e^{2x} 5 e 2 x , try A e 2 x Ae^{2x} A e 2 x
Product of functions requires product of individual particular solutions
Example: For x e x xe^x x e x , try A x e x Axe^x A x e x
Trigonometric and Special Cases
Trigonometric nonhomogeneous terms (sin(bx) or cos(bx)) use form A sin ( b x ) + B cos ( b x ) A \sin(bx) + B \cos(bx) A sin ( b x ) + B cos ( b x )
Example: For 3 sin ( 4 x ) 3\sin(4x) 3 sin ( 4 x ) , try A sin ( 4 x ) + B cos ( 4 x ) A\sin(4x) + B\cos(4x) A sin ( 4 x ) + B cos ( 4 x )
When nonhomogeneous term matches homogeneous solution, multiply by x k x^k x k (k is root multiplicity)
Example: If e x e^x e x is both nonhomogeneous term and homogeneous solution, try A x e x Axe^x A x e x
Method of undetermined coefficients applies to combinations of polynomials, exponentials, sines, and cosines
Example: For 2 x + 3 e x + sin ( x ) 2x + 3e^x + \sin(x) 2 x + 3 e x + sin ( x ) , try A x + B + C e x + D sin ( x ) + E cos ( x ) Ax + B + Ce^x + D\sin(x) + E\cos(x) A x + B + C e x + D sin ( x ) + E cos ( x )
Undetermined Coefficients Method
Procedure and Implementation
Assume particular solution based on nonhomogeneous term form
Substitute assumed solution into original differential equation
Collect like terms and equate coefficients to solve for undetermined constants
Solve resulting system of algebraic equations for coefficient values
Method most effective for constant coefficient equations with specific nonhomogeneous terms
Example: Solve y ′ ′ + 4 y = x 2 y'' + 4y = x^2 y ′′ + 4 y = x 2
Assume y p = A x 2 + B x + C y_p = Ax^2 + Bx + C y p = A x 2 + B x + C
Substitute: ( 2 A ) x 0 + ( 2 B ) x 1 + ( A x 2 + B x + C ) = x 2 (2A)x^0 + (2B)x^1 + (Ax^2 + Bx + C) = x^2 ( 2 A ) x 0 + ( 2 B ) x 1 + ( A x 2 + B x + C ) = x 2
Equate coefficients: A = 1 / 4 A = 1/4 A = 1/4 , B = 0 B = 0 B = 0 , C = − 1 / 8 C = -1/8 C = − 1/8
Particular solution: y p = 1 4 x 2 − 1 8 y_p = \frac{1}{4}x^2 - \frac{1}{8} y p = 4 1 x 2 − 8 1
Verification and Special Considerations
Verify particular solution by substituting back into original equation
Take special care when assumed form matches homogeneous solution
Multiply by x k x^k x k in these cases
Example: For y ′ ′ − y = e x y'' - y = e^x y ′′ − y = e x , try y p = A x e x y_p = Axe^x y p = A x e x instead of y p = A e x y_p = Ae^x y p = A e x
Method limitations include complex nonhomogeneous terms and variable coefficient equations
Example: y ′ ′ + x y = ln ( x ) y'' + xy = \ln(x) y ′′ + x y = ln ( x ) requires different approach (variation of parameters)
General Solution of Nonhomogeneous Equations
Combining Solutions
General solution combines complementary (homogeneous) and particular solutions
Complementary solution from associated homogeneous equation (using characteristic equations for constant coefficients)
Example: For y ′ ′ + 4 y = 0 y'' + 4y = 0 y ′′ + 4 y = 0 , complementary solution y c = c 1 cos ( 2 x ) + c 2 sin ( 2 x ) y_c = c_1\cos(2x) + c_2\sin(2x) y c = c 1 cos ( 2 x ) + c 2 sin ( 2 x )
Particular solution found using undetermined coefficients or variation of parameters
General solution contains arbitrary constants from complementary solution
Example: y = c 1 cos ( 2 x ) + c 2 sin ( 2 x ) + 1 4 x 2 − 1 8 y = c_1\cos(2x) + c_2\sin(2x) + \frac{1}{4}x^2 - \frac{1}{8} y = c 1 cos ( 2 x ) + c 2 sin ( 2 x ) + 4 1 x 2 − 8 1 (combining previous examples)
Verification and Applications
Verify general solution satisfies original equation through substitution and differentiation
Apply principle of superposition for equations with multiple nonhomogeneous terms
Example: For y ′ ′ + y = sin ( x ) + e x y'' + y = \sin(x) + e^x y ′′ + y = sin ( x ) + e x , find particular solutions for each term separately and add
Use general solution to satisfy initial or boundary conditions
Example: Solve y ( 0 ) = 1 y(0) = 1 y ( 0 ) = 1 , y ′ ( 0 ) = 0 y'(0) = 0 y ′ ( 0 ) = 0 using general solution to determine c 1 c_1 c 1 and c 2 c_2 c 2
Variation of Parameters Method
Method Overview and Setup
General method for finding particular solutions of nonhomogeneous linear equations
Applicable to wider range of nonhomogeneous terms than undetermined coefficients
Assumes particular solution form y p = u 1 y 1 + u 2 y 2 y_p = u_1y_1 + u_2y_2 y p = u 1 y 1 + u 2 y 2
y 1 y_1 y 1 and y 2 y_2 y 2 are linearly independent homogeneous solutions
u 1 u_1 u 1 and u 2 u_2 u 2 are functions to be determined
Requires knowledge of general homogeneous solution
Example: For y ′ ′ + y = sec ( x ) y'' + y = \sec(x) y ′′ + y = sec ( x ) , use y 1 = cos ( x ) y_1 = \cos(x) y 1 = cos ( x ) and y 2 = sin ( x ) y_2 = \sin(x) y 2 = sin ( x )
Implementation and Calculations
Solve system of first-order differential equations for u 1 ′ u_1' u 1 ′ and u 2 ′ u_2' u 2 ′
Integrate to find u 1 u_1 u 1 and u 2 u_2 u 2
Wronskian of homogeneous solutions crucial in method (denominator of u 1 ′ u_1' u 1 ′ and u 2 ′ u_2' u 2 ′ expressions)
Example: Wronskian W = y 1 y 2 ′ − y 2 y 1 ′ W = y_1y_2' - y_2y_1' W = y 1 y 2 ′ − y 2 y 1 ′
Extend to higher-order equations with additional terms in particular solution
Example: For third-order, use y p = u 1 y 1 + u 2 y 2 + u 3 y 3 y_p = u_1y_1 + u_2y_2 + u_3y_3 y p = u 1 y 1 + u 2 y 2 + u 3 y 3
Advantages and Limitations
More general than undetermined coefficients, applicable to complex nonhomogeneous terms
Example: Can solve y ′ ′ + y = tan ( x ) y'' + y = \tan(x) y ′′ + y = tan ( x ) where undetermined coefficients fails
Often leads to more complex integrations
May be computationally intensive for certain nonhomogeneous terms
Example: y ′ ′ + y = e x 2 y'' + y = e^{x^2} y ′′ + y = e x 2 requires challenging integrations
Useful when undetermined coefficients is not applicable or efficient
Example: Effective for equations with variable coefficients like x y ′ ′ + y = ln ( x ) xy'' + y = \ln(x) x y ′′ + y = ln ( x )