how to find explicit solution

**lord12** · September 19th 2009, 06:52 PM

(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"?

**Prove It** · September 19th 2009, 07:34 PM

Originally Posted by lord12

(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"?

This is a separable DE

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

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

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

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

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

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

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

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

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

Now use the initial condition to find C.

**lord12** · September 20th 2009, 10:10 AM

but ln(-1) is undefined!

**mr fantastic** · September 20th 2009, 03:20 PM

Originally Posted by lord12

but ln(-1) is undefined!

No. It isn't. (For real numbers, anyway).

Luckily however, the solution you were given does not lead to this number .... (In other words, read the given answer more carefully).

**Prove It** · September 20th 2009, 07:36 PM

Originally Posted by lord12

but ln(-1) is undefined!

Does the answer ask you to find $\ln{(-1)}$ ?

No.

It asks you to find $\ln{|-1|} = \ln{1} = 0$ .

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