Conversation started Sep 20, 2012 at 22:50.
hhh
hhh
Sep 20, 2012 22:50
Suppose a ODE such as $\dot y = x$, is it hard to check whether this is linear or not? $f(ax)=a f(x)$ and $f(a+b)=f(a)+f(b)$.
Now $y(x)=x^2 /2+C$ where $C\in\mathbb R$.
$y(ax)=(a^2/2) x^2 + C$ and $a f(x) = a( x^2 /2 + C)$.
$y_2(ax)=(a^2/2) x^2 + aC$
$y_1(ax)=(a^2/2) x^2 + C$
Evan
Evan
Have any of you used MyMathLab?
hhh
hhh
But now $y_2=y_1$ for some arbitrary constants $C_1, C_2 \in \mathbb R$, right?
So this ODE $\dot y =x$ is linear.
ERR
$a y_2(x)=(a^2/2) x^2 + aC_2$
$y_1(ax)=(a^2/2) x^2 + C_1$
Now $a y_2(x) = y_1(a x)$ for $C_1, C_2 \in \mathbb R$.
where $y_1 = y_2$.
- so one of the linear property holds.
Jeff
Jeff
Sep 20, 2012 23:06
@Evan i have
hhh
hhh
$f(a+b)=a^2/2+ab+b^2/2+C_3$
$f(a)+f(b)=a^2/2+b^2/2+C_4$
So $g(a,b) := f(a+b)-(f(a)+f(b))=ab+C_3-C_4$.
Because $g(a,b)\not =0$ for every $a,b\in\mathbb R$, $f$ is not linear, correct?
 
Conversation ended Sep 20, 2012 at 23:10.