FODE_25-27

FODE_25-27 - Tutorial FODE: Ordinary First-Order...

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Tutorial FODE: Ordinary First-Order Differential Equations b. The equation is scale invariant for n = ¡ 1 = 2 = ) y = v= p x = ) ¡ v 2 +1 ¢ dv v ( v 2 ¡ 1) = 3 2 dx x : A second change of variable, z = v 2 ; gives μ 2 z ¡ 1 ¡ 1 z dz = 3 dx x ; which integrates to give the implicit expression ( z ¡ 1) 2 = Czx 3 ; where z = xy 2 : c. Note that the exponent must be scale invariant, giving n =1 = 2 : This sets the scale that the restof the termsmustpossess. Since theyaremanifestlyscale-invariant with respect to each other, both beingof order xy; anyvalue of n woulddowere the exponentialnotpresent. We make the variable change y = vx 1 = 2 = ) e v 2 (2 xdv + vdx ) ¡ 3 vdx =0 = ) 2 dv v (1 ¡ 3 e ¡ v 2 ) = ¡ dx x = ) 2 Z v dv 0 v 0 (1 ¡ 3 e ¡ v 0 2 ) = ln( C=x ) : Exercise 25. a. @A @x = @ (2 y +3 x ) =3; @B @y = @ ¡ 3 x 2 y ¢ = 3 : The equation is therefore exact. We can therefore compute u ( x; y ) thusly: u ( x; y ) = Z y ¡ 2 y 0 x ¢ dy 0 = y 2 xy + g ( x ) ; u ( x; y ) = Z x ¡ 3 x 0 2 y ¢ dx 0 = x 3 xy + h ( y )= ) u ( x; y ) = x 3 xy + y 2 : The solution is implicit in theequation x 3 xy + y 2 = C: For y (0) =1 ; we have C : b.
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This note was uploaded on 07/13/2008 for the course PHY 201 taught by Professor Covatto during the Spring '08 term at ASU.

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FODE_25-27 - Tutorial FODE: Ordinary First-Order...

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