DE - Reduction of Order questions
- From: "Nick" <chanlikhangnick@xxxxxxxxx>
- Date: 12 Aug 2006 22:17:07 -0700
1. Verify that u_1 = sin (x^2) is a solution of x u'' - u' + 4 x^3 u =
0 and use reduction of order to find a second, linearly independent
solution.
2. Verify that u_1 = x + 1 is a solution of x u'' - ( x + 1 ) u' + u =
0 and use reductio of order to find the general solution.
1. I've verified u_1 and tried u_2 = v (x) u_1 as another solution.
Substituting and rearranging terms gives x v'' + ( 4 x^2 cos (x^2) -
sin (x^2) ) v' = 0. Then let w = v', we have x w' + ( 4 x^2 cos (x^2) -
sin (x^2) ) w = 0, which is a first order, linear and homogeneous ODE.
Did I make any mistake so far? If not, I proceeded and found difficulty
in integrating [sin (x^2)] / x.
2. The reduced equation I found was ( x + 1 ) v'' + [ 2 - ( x + 1 )^2 ]
v' = 0. Did I make any mistake? I found difficulty in solving the ODE
again.
Answer
1: cos (x^2)
2: A exp(x) + B ( x + 1 )
I'm taking my first Differential Equation course.
Regards
Nick
.
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