(1) Consider the initial value problem

dx/dt = -1/tx

, x(−1) = 1

(a) Use an analytic method to ﬁnd an explicit solution for this

problem. What is the domain of deﬁnition for this solution?

What does it mean to "use an analytic method"?

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- Sep 19th 2009, 06:52 PMlord12how to find explicit solution
(1) Consider the initial value problem

dx/dt = -1/tx

, x(−1) = 1

(a) Use an analytic method to ﬁnd an explicit solution for this

problem. What is the domain of deﬁnition for this solution?

What does it mean to "use an analytic method"? - Sep 19th 2009, 07:34 PMProve It
This is a separable DE

$\displaystyle \frac{dx}{dt} = -\frac{1}{tx}$

$\displaystyle x\,\frac{dx}{dt} = -\frac{1}{t}$

$\displaystyle \int{x\,\frac{dx}{dt}\,dt} = \int{-\frac{1}{t}\,dt}$

$\displaystyle \int{x\,dx} = -\ln{|t|} + C_1$

$\displaystyle \frac{1}{2}x^2 + C_2 = -\ln{|t|} + C_1$

$\displaystyle \frac{1}{2}x^2 = -\ln{|t|} + C_1 - C_2$

$\displaystyle x^2 = -2\ln{|t|} + 2C_1 - 2C_2$

$\displaystyle x^2 = -2\ln{|t|} + C$, where $\displaystyle C = 2C_1 - 2C_2$

$\displaystyle x = \sqrt{C - 2\ln |t|}$

Now use the initial condition to find C. - Sep 20th 2009, 10:10 AMlord12
but ln(-1) is undefined!

- Sep 20th 2009, 03:20 PMmr fantastic
- Sep 20th 2009, 07:36 PMProve It