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Answer Guide 4 Math 427K: Unique Number 55160 Wednesday, September 21, 2011 1. (a) Find the Wronskian W ( t ) of u 1 ( t ) = e ° 4 t and u 2 ( t ) = te ° 4 t . Do u 1 and u 2 form a fundamental set of solutions to u 00 + 8 u 0 + 16 u = 0 ? Solution: We °rst verify by substitution that they are solutions. Then we compute W [ u 1 ; u 2 ]( t ) = det ° e ° 4 t te ° 4 t ° 4 e ° 4 t e ° 4 t (1 ° 4 t ) ± = e ° 8 t : ( F ) Because e ° 8 t never vanishes, they do form a fundamental set of solutions. (b) If the Wronskian of u ( t ) = e 3 t and v ( t ) is W ( t ) = 6 t , what are all possible choices for v ( t ) ? Solution: We compute 6 t = W [ u; v ]( t ) = det ° e 3 t v 3 e 3 t v 0 ± = e 3 t ( v 0 ° 3 v ) : So v solves the linear °rst-order ode v 0 ° 3 v = 6 te ° 3 t , whose general solution is v ( t ) = Ae 3 t ° ( 1 6 + t ) e ° 3 t : ( F ) 2. (a) Show that u 1 ( t ) = t cos t and u 2 ( t ) = t sin t are both solutions of the linear homoge- neous equation u 00 ° 2 t u 0 + (1 + 2 t 2 ) u = 0 ( t > 0) : Then compute their Wronskian. Do they form a fundamental set of solutions? In which domain? Solution: A direct calculation (omitted here) veri°es that u 1 and u 2 are both solu- tions. We compute W [ u 1 ; u 2 ]( t ) = det ° t cos t t sin t cos t ° t sin t sin t + t cos t ± = t 2 : (1) Recall that Abel±s theorem tells us that the Wronskian is either always zero or never zero in the largest domain where the coe¢ cients of the di/erential equation are contin- uous. So u 1 ; u 2 form a fundamental set of solutions in the given domain t > 0 . (Note that t < 0 would also work if that had been de°ned as the domain.)

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(b) Show that y 1 ( x ) = e cos x and y 2 ( x ) = e ° cos x are both solutions of the linear homoge- neous equation y 00 ° (cot x ) y 0 ° (sin x ) 2 y = 0 (0 < x < ° ) : Then compute their Wronskian. Do they form a fundamental set of solutions?
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