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178_pdfsam_math 54 differential equation solutions odd

# 178_pdfsam_math 54 differential equation solutions odd -...

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Chapter 4 For t = 0, y 1 (0) = 3 · 0 2 = 0 and y 2 (0) = 0. The latter follows from the fact that one-sided derivatives of y 2 ( t ), 3 t 2 and 3 t 2 , are both zero at t = 0. Also, y 1 (0) = y 2 (0) = 0. Hence W [ y 1 , y 2 ](0) = 0 0 0 0 = 0 , and so W [ y 1 , y 2 ]( t ) 0 on ( −∞ , ). This result does not contradict part (b) in Prob- lem 34 because these functions are not a pair of solutions to a homogeneous linear equation with constant coeﬃcients. 37. If y 1 ( t ) and y 2 ( t ) are solutions to the equation ay + by + c = 0, then, by Abel’s formula, W [ y 1 , y 2 ]( t ) = Ce bt/a , where C is a constant depending on y 1 and y 2 . Thus, if C = 0, then W [ y 1 , y 2 ]( t ) = 0 for any t in ( −∞ , ), because the exponential function, e bt/a , is never zero. For C = 0, W [ y 1 , y 2 ]( t ) 0 on ( −∞ , ). 39. (a) A linear combination of y 1 ( t ) = 1, y 2 ( t ) = t , and y 3 ( t ) = t 2 , C 1 · 1 + C 2 · t + C 3 · t 2 = C 1 + C 2 t + C 3 t 2 , is a polynomial of degree at most two and so can have at most two real roots, unless it is a zero polynomial, i.e., has all zero coeﬃcients. Therefore, the above linear combination vanishes on (
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