Transcription of Solutions of Linear Differential Equations
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Solutions of Linear Differential Equations
Appendix A Solutions of Linear Differential Equations Linear Differential Equations with Constant Coefficients Linear diflFerential Equations with constant coefficients are usually writ-ten as 2/("> + ai2/("-i) + .. + a _i2/(i) + anV = g, ( ) where a^, fc = 1,.., n, are numbers, y^^^ = ^ , and g = g{t) is a known function of t. We shall denote hy D = ^ the derivative operator^ so that the Differential equation now becomes p{D)y = (D^ + aiD^-i + .. + a^_iD + an)y = g. ( ) If g(t) = 0, the equation is said to be homogeneous. If g{t) ^ 0, then the homogeneous or reduced equation is obtained from ( ) by replacing g byO.}}}
A, 7. Reduction of Higher-Order to First-Order Linear Equations 369 A.7 Reduction of Higher-Order Linear Equations to Systems of First-Order Linear Equations Another way of solving equation (A.l) is to convert it into a system of first-order linear equations. We use the transformations zi = y, Z2 = y^^\...,zn = y^'' ^\ (A.8)
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