Question 1145007
<pre>
{{{dx/dt = t(x - 2)}}}

Multiply both sides by dt:

{{{dx = t(x - 2)*dt}}}

The term on the right side contains two variables, t and x. We must get rid
of the other variable besides the variable of the differential.  The variable
of the differential on the right side is t (because we have dt), so we must
get rid of the (x - 2) factor.  So we divide both sides by (x - 2):

{{{dx/(x-2) = (t(x - 2)*dt)/(x-2)}}}

{{{dx/(x-2) = (t(cross(x - 2))*dt)/(cross(x-2))}}}

{{{dx/(x-2) = t*dt)}}}

We integrate both sides:

{{{int(dx/(x-2))}}}{{{""=""}}}{{{int(t^""*dt))}}}

We use the formula {{{int(du/u)}}}{{{""=""}}}{{{"ln|u|"+C}}} on the left side,
and we use the formula {{{int(u^n*du)}}}{{{""=""}}}{{{u^(n+1)/(n+1)+C}}} on the right side:

{{{ln(abs(x-2))=t^2/2^""+C[1]}}}

{{{matrix(2,1,"",e^(t^2+C[1])=abs(x-2))}}}


Since the left side is positive, we can dispense with the absolute value on the
right side and require that x<u>></u>2.

{{{matrix(2,1,"",e^(t^2+C[1])=x-2)}}}

{{{matrix(2,1,"",e^t^2*e^C[1]=x-2)}}}

Since e<sup>C<sub>1</sub></sup> is an arbitrary constant, we can let it be C.

{{{matrix(2,1,"",Ce^t^2=x-2)}}}

{{{matrix(2,1,"",x=Ce^t^2+2)}}}

Edwin</pre>