My Math Forum Urgent! question about differentials

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 November 18th, 2011, 01:39 PM #1 Member   Joined: Nov 2010 Posts: 53 Thanks: 0 Urgent! question about differentials Hi, a friend of mine asked me to help him with some math problems, but it has been ages since I did this kind of stuff. Could anyone help me to solve this problem: Find the time required for a sum of money to double itself at 5% per annum compunded continuosly. Hint. dx/dt = 0.05x, where x is the amount after t years.
 November 18th, 2011, 01:49 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Urgent! question about differentials Essentially, we need to solve the IVP: $\frac{dx}{dt}=0.05x$ where $x$$0$$=x_0$ The ODE is separable: $\frac{dx}{x}=0.05\,dt$ Integrate, using initial and final conditions as boundaries: $\int_{x_0}\,^x\frac{dx}{x}=0.05\int_0\,^t\,dt$ $\ln$$\frac{x}{x_0}$$=0.05t$ Now, to find the doubling time we set $x=2x_0$ to get: $\ln(2)=0.05t$ Solve for t: $t=20\ln(2)\approx13.86$ Thus, at 5% annual interest compounded continuously, it takes approximately 13.86 years for the investment to double in value.
 November 19th, 2011, 12:54 AM #3 Member   Joined: Nov 2010 Posts: 53 Thanks: 0 Re: Urgent! question about differentials Thanks a lot for your answer, nice and thorough.

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