(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"?
(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
$\displaystyle \frac{dx}{dt} = -\frac{1}{tx}$
$\displaystyle x\,\frac{dx}{dt} = -\frac{1}{t}$
$\displaystyle \int{x\,\frac{dx}{dt}\,dt} = \int{-\frac{1}{t}\,dt}$
$\displaystyle \int{x\,dx} = -\ln{|t|} + C_1$
$\displaystyle \frac{1}{2}x^2 + C_2 = -\ln{|t|} + C_1$
$\displaystyle \frac{1}{2}x^2 = -\ln{|t|} + C_1 - C_2$
$\displaystyle x^2 = -2\ln{|t|} + 2C_1 - 2C_2$
$\displaystyle x^2 = -2\ln{|t|} + C$, where $\displaystyle C = 2C_1 - 2C_2$
$\displaystyle x = \sqrt{C - 2\ln |t|}$
Now use the initial condition to find C.