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20b(4b3)3
What is a solution to the differential equation dydx=2csc2xcotx?
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Does the logarithm inequality extend to the complex plane?For estimates, the inequality log ( y ) ≤ y − 1 , y > 0 is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, | log ( z ) | ≤ something for all z ∈ C Perhaps this would work?: log ( z ) ≤ log 2 | z | + arg ( z ) 2
Entropy change in a calorimetry problemA standard textbook problem has us calculate the change in entropy in a system that undergoes some sort of heat exchange. For example, object A has specific heat c a and initial temperature T A and object B has specific heat c b with initial temperature T B . They are they put in contact with each other until they reach thermal equilibrium, and our goal is to find the total entropy change of the system.The standard solution is to use S = ∫ d Q T where d Q = m c d T. But the above integral is only satisfied for reversible processes, whereas this heat exchange is clearly irreversible.The usual workaround for this is to pick some reversible path and calculate the entropy change on our "fake" path, since entropy is a state variable. For example, in the free expansion of an ideal gas, we pick calculate the entropy change along an isotherm that carries us along the expansion to find the true change in entropy.My question is - what exactly is the reversible path we are using when we use d Q = m c d T?
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How can I show that the Euler method fails to approximate the exact solution y ( x ) = ( 2 x / 3 ) 3 / 2 to the IVP y ′ = y 1 3 y ( 0 ) = 0Here we have f ( t , y ) = y 1 3 , y 0 = 0 and so f ( t 0 , y 0 ) = f ( 0 , 0 ) = 0 and y n + 1 = y n + h f ( t n , y n )Thus y 1 = 0 y 2 = 0 ⋮ y n = 0So, I can't understand why it fails. Could you help me?
Let f : R → R be a continuous function and g : R → R be a Lipschitz function. Would you help me to prove that the system of differential equation x ′ = g ( x ) y ′ = f ( x ) ywith initial value x ( t 0 ) = x 0 and y ( t 0 ) = y 0 has a unique solution.Could I prove the uniqueness solution of x ′ = g ( x ), x ( t 0 ) = x 0 by Gronwall Inequality first then use the result to prove the second?
If the log ( x ) = − 7.65, how do you find x?
Use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. ∑n=1∞1n3+2n
If I toss a coin 3 times and want to know the probability of at least one head, I have understood that the answer is 1 − 0.5 3 = 99 %. However, why cannot I not use the additon rule P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ), i.e. 0.5 + 0.5 + 0.5 − 0.5 3 ?
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