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a. show that if u1is a solution of the third -order variable coefficient linear differential equation u'''+p1(x)u''+p2(x)u'+p3(x)u=0 then the substitution u(x)=u1(x)v(x)leads to the second order differential equation for v' u1v'''+(3u1'+p1u1)v''+(3u1''+2p1u1'+p2u1)=0
b. verify that u1(x)=e^x is a solutions of the differential equation (2-x)u'''+(2x-3)u''-xu'+u=0 use the method in part (a) to find the general solution of the differential equation.
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u(x)=u1(x)v(x)
u = u1 * v ................(微分 = 前項微分 + 尾相微分)
u' = u1' * v + u1 * v '
u'' = (u1'' * v + u1' * v ') + (u1' * v ' + u1 * v '') = u1'' * v + 2 (u1' * v') + u1 * v ''
u''' = u1''' * v + 3 (u1'' * v ') + 3 (u1' * v '') + u1 * v '''
帶入 u'''+p1(x)u''+p2(x)u'+p3(x)u=0 即可得證
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(2-x)u'''+(2x-3)u''-xu'+u=0 全除 (2-x)
與u'''+p1(x)u''+p2(x)u'+p3(x)u=0 比較知p1.. p2.. p3
又以知 u = u1 * v = e^x * v 仿照a的結果可得到答案