# behavior of solutions

• Oct 13th 2009, 02:16 PM
elmo
behavior of solutions
If a>0, b>0, c=0, show that all solutions of ay''+by'+cy=0 approach a constant that depends on the initial conditions as t approaches to infinity. Determine this constant for the initial conditons $\displaystyle y(0)=y_0, y'(0)=y_0'$
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my approach:
$\displaystyle y=c_1e^{r_1 t}+c_2e^{r_2 t}$
$\displaystyle b^2-4ac=b^2$
$\displaystyle r_1=0, r_2= \frac{-b-b}{2a}$
$\displaystyle y=c_1e^0+c_2e^{\frac{-bt}{a}}$
$\displaystyle y(0)=c_1=y_0$
$\displaystyle y'(t)=c_2(-{\frac{b}{a})e^{\frac{-bt}{a}}}$
$\displaystyle t=0, c_2=y_0'(-{\frac{a}{b}})$
$\displaystyle y=c_1e^0+c_2e^{\frac{-bt}{a}}$
$\displaystyle =c_1+c_2e^{\frac{-bt}{a}}$
$\displaystyle =y_0+{y_0'}(-{\frac{a}{b}})e^{\frac{-bt}{a}}$
which approaches to $\displaystyle y_0$ as t approaches to infinity.

(according to the solution manual, the constant should be $\displaystyle y_0+{\frac{a}{b}}y_0'$ )

• Oct 14th 2009, 06:06 AM
Hello elmo
Quote:

Originally Posted by elmo
If a>0, b>0, c=0, show that all solutions of ay''+by'+cy=0 approach a constant that depends on the initial conditions as t approaches to infinity. Determine this constant for the initial conditons $\displaystyle y(0)=y_0, y'(0)=y_0'$
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my approach:
$\displaystyle y=c_1e^{r_1 t}+c_2e^{r_2 t}$
$\displaystyle b^2-4ac=b^2$
$\displaystyle r_1=0, r_2= \frac{-b-b}{2a}$
$\displaystyle y=c_1e^0+c_2e^{\frac{-bt}{a}}$
$\displaystyle \color{red}y(0)=c_1=y_0$
$\displaystyle y'(t)=c_2(-{\frac{b}{a})e^{\frac{-bt}{a}}}$
$\displaystyle t=0, c_2=y_0'(-{\frac{a}{b}})$
$\displaystyle y=c_1e^0+c_2e^{\frac{-bt}{a}}$
$\displaystyle =c_1+c_2e^{\frac{-bt}{a}}$
$\displaystyle =y_0+{y_0'}(-{\frac{a}{b}})e^{\frac{-bt}{a}}$
which approaches to $\displaystyle y_0$ as t approaches to infinity.

(according to the solution manual, the constant should be $\displaystyle y_0+{\frac{a}{b}}y_0'$ )

See the line I've highlighted in red. Can you see your mistake? $\displaystyle e^0=1$, not $\displaystyle 0$.