AM105b,Spring 2005 FIRST AND SECOND ORDER ODEs—ESSENTIALS General solution to linear first order ode:( )( ) ya x yh x′ +=1( )( )( )P y xcyxyx =+withhomogeneous solution0 ( )1 ( )x x adyxe xx -∫ =andparticular solution1 1 1 ( )( )( )( )x Px hyxyxdyx x x =∫ wherec,0xand1 xare constants that can be chosen at the discretion of the analyst. General solution to constant coefficient linear second order ode: ( ) yaybyh x′′′++=1122( )( )( )( )P y xc yxc yxyx =++Homogeneous solutions—three possibilities, depending onaand b If2/ 4ab, 12 12,xxyeyemm==with2212 / 2/ 4,/ 2/ 4aabaab mm= -+-= ---(Note that this includes the case for0b=, with12 1,axyye-==) If2/ 4ab<, 12 sin,cosxxyex yex mmnn==with2/2,/ 4 abamn= -=-(Note that this includes the case for0a=, with12 sin,cosyx yxnn==) If2/ 4ab=, the method of reduction of order (see below) can be used to show 12 ,xxyeyxemm==with/ 2 am= -(Note that this includes the case for0ab==, with12 1,yyx==) Particular solution
