AMath350.W13.A3.Sol

# The equation dx d 2x can be rewritten as e y ex and

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Unformatted text preview: be rewritten as e y = ex , and so we ﬁnd that dx a) For y=e 2x (ex + C ) = e x + Ce 2x . We can see that y = e x is an exceptional solution, and all other solutions approach it as x ! 1, and are repelled by it as x ! 1. Next, observe that y 0 = 0 if and only if y = 1 e x (this is our horizontal 2 isocline). The solutions must look something like this: AMATH 350 Assignment #3 - Winter 2013 Page 3 dy b) For x = 2y + x3 , we need to start by putting the equation in standard dx form: dy 2 y = x2 . dx x R 2 An integrating factor is µ(x) = e x dx = e 2 ln x = 1/x2 . Incorporating this, we obtain 1 dy 2 y=1 2 dx x x3 which must be the same equation as Therefore and so dy ⇣ y ⌘ = 1. dx x2 y =x+C x2 y = x3 + Cx2 . Now, y = x3 is an exceptional solution. All other solutions approach this as x ! 0 and diverge from it as x ! ±1. x3 The horizontal isocline is y = . Finally, it might help to recognize that 2 every solution passes through the origin. This is unusual, since solution curves usually cannot cross, but in this particular example the conditions of the Existence and Uniqueness Theorem are not met on the y -axis. Putting our observations together, we arrive at the following sketch: AMATH 350 Assignment #3 - Winter 2013 Page 4 3. (Interest Rate Problem) a) An initial value problem for the interest rate is dr = 0.0015, r(0) = 0.05 dt Since y = xv , the general solution of the original DE is x Solve this y = ln |the+ C. + ln to ﬁnd x| variable interest rate r(t). y Solution: x dr the original r(0) = 0.05 . 6. (a) Sincefor = xv , the general solution of = 0.0015 DE is IVP y interest rate r(t): x y dt + ln = ln |x| + C. initial rate is 5% y xZ rate decreases 0.15% per year Z Solve DE: dr = 0.0015 dt )dr = 0.0015t + C . r = 0.0015 r(0) = 0.05 . 6. (a) IVP for interest rate r(t): Apply IC: r(0) = 0.05 ) C = 0.05. dt initial rate is 5% Interest rate at time t: r(t) = decreases0015t. per year rate 0.05 0. 0.15% Z Z Since (t= solve general0015 investment 0(dollars) is the (b) Set upyandxv ,dr =value 0ofsolutionproblem for theDE at. time t investment, Let V ) . va...
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