2.1we - 2.1 FIRST-ORDER LINEAR EQUATIONS KIAM HEONG KWA An...

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2.1: FIRST-ORDER LINEAR EQUATIONS KIAM HEONG KWA An n th order ODE is called linear if one can write it in the form (1) a 0 ( t ) y ( n ) + a 1 ( t ) y ( n - 1) + · · · + a n - 1 ( t ) y 0 + a n ( t ) y = g ( t ) , where the coefficients a i ( t ) ( i = 0 , 1 , · · · , n ) and g ( t ) are known func- tions; otherwise, the equation is called nonlinear . In particular, a first- order linear equation is an equation of the form (2) dy dt + p ( t ) y = g ( t ) , where p ( t ) and g ( t ) are given continuous functions on some open inter- val I . Theorem 1. Let p ( t ) and q ( t ) be continuous on an open interval I R that contains a value t 0 of the independent variable t . Then for any prescribed value y 0 , the IVP consisting of (2) and the initial condition y ( t 0 ) = y 0 has a unique solution on I . Proof of existence. The solution to the IVP problem is constructed us- ing the method of integrating factors as follows. First, one multiplies (2) by a nonzero function μ ( t ) to be determined to yield (3) μ ( t ) dy dt + μ ( t ) p ( t ) y = μ ( t ) g ( t ) .
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