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Unformatted text preview: Fall 2003 Math 308/501502 2 First-Order Differential Equations 2.3 Linear Equations Wed, 10/Sep c 2003, Art Belmonte Summary We saw in 1 . 1 that a first-order linear differential equation has the form a 1 ( x ) dy dx + a ( x ) y = b ( x ) . The coefficients a , a 1 , b are constants or functions of the independent variable x alone; they do not depend on the dependent variable y . Moreover, note that both y and its derivative y occur to the first power only. The equation is homogeneous if b ( x ) = 0 and nonhomogeneous if b ( x ) 6= 0. Conventional procedure (CP) 1. You MUST put the DE in standard linear form ( SLF ): y + P ( x ) y = Q ( x ) 2. Construct an integrating factor ( x ) = exp ( R P ( x ) dx ) . Note that any antiderivative R p ( x ) dx will do. So take the constant of integration to be zero. 3. Multiply the SLF by the integrating factor to obtain y + Py = Q . Notice by design that ( y ) = y + y = y + Py (via the Chain Rule). Putting these two together gives (( x ) y ( x )) = ( x ) Q ( x ) which is separable. That is, its of the form d w/ dx = g ( x ) and we may use 2 . 2 techniques!...
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