- #1
Nickg140143
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Homework Statement
Verify that the indicated funciton is a solution of the given Differential Equation. c1 and c2 denote constants where appropriate.
[tex]\frac { dX }{ dt } =(2-x)(1-x);\quad \quad \ln { \frac { 2-x }{ 1-x } } =t[/tex]
The Attempt at a Solution
I'm not quite sure how to really start this problem. If I'm reading the question right, the differential equation is
[tex]\frac { dX }{ dt } =(2-x)(1-x)[/tex]
and that the solution I'm checking is
[tex]\ln { \frac { 2-x }{ 1-x } } =t[/tex]
I was thinking of perhaps integrating
[tex]\frac { dX }{ dt } =(2-x)(1-x)[/tex]
then maybe plug in the the given value of t?
I've read through the section and looked at my notes from class, but I can't seem to fully understand what I should be doing in this problem. The solution in the back of the book says
[tex]\frac { d }{ dt } \ln { \frac { 2-X }{ 1-X } } =1,\quad \left[ \frac { -1 }{ 2-X } +\frac { 1 }{ 1-X } \right] \frac { dX }{ dt } =1[/tex]
Simplifies to
[tex]\frac { dX }{ dt } =(2-X)(1-X)[/tex]
...Not entirely sure what it necessarily means by that though. Any help would be GREATLY appreciated.