Theoretical Ordinary Differential Equation with the lim Yg = 0
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I was given a theoretical problem in my Linear Algebra and Differentials class and I am stumped on how to approach this problem.
Consider the ODE y''+ay'+by=0 where a and b are constants such that $a^2$ < 4b. What conditions on a and b guarentees that the general solution of the ODE satisfies $displaystyle{lim_{x to infty} Yg=0}$
So I do understand that Yg is the general solution, which contains the homogeneous solution + the particular solution.
So far we have worked with simple polynomials in our class, so what we do is we solve for the particular solution first and use $y=C1e^(r1*x) + C2e^(r2*x)$ where t1 and t2 are the roots to the equation $r^2+ar+b=0$ (from above equation).
The way we solve the particular solution is by "Guessing" the right hand side of this equation, but since it is = 0 I believe that the Yg is just the Y homogeneous.
$Yg = Yh+Yp => Yg = Yh+0 => Yg=Yh$
So in this case I believe my Yg would be of the form $Yg = e^(r1*x) + e^(r2*x)$ which implies that my lim equation is $displaystyle{lim_{x to infty} e^(r1*x) + e^(r2*x)=0}$
I am not sure if this is right at all, but this is how I would currently approach it. I just have no idea how to satisfy the constraint given of $a^2<4b$ and that the limit of Yg is = 0.
Any guidance would be greatly appreciated.
ordinary-differential-equations limits
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
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add a comment |
$begingroup$
I was given a theoretical problem in my Linear Algebra and Differentials class and I am stumped on how to approach this problem.
Consider the ODE y''+ay'+by=0 where a and b are constants such that $a^2$ < 4b. What conditions on a and b guarentees that the general solution of the ODE satisfies $displaystyle{lim_{x to infty} Yg=0}$
So I do understand that Yg is the general solution, which contains the homogeneous solution + the particular solution.
So far we have worked with simple polynomials in our class, so what we do is we solve for the particular solution first and use $y=C1e^(r1*x) + C2e^(r2*x)$ where t1 and t2 are the roots to the equation $r^2+ar+b=0$ (from above equation).
The way we solve the particular solution is by "Guessing" the right hand side of this equation, but since it is = 0 I believe that the Yg is just the Y homogeneous.
$Yg = Yh+Yp => Yg = Yh+0 => Yg=Yh$
So in this case I believe my Yg would be of the form $Yg = e^(r1*x) + e^(r2*x)$ which implies that my lim equation is $displaystyle{lim_{x to infty} e^(r1*x) + e^(r2*x)=0}$
I am not sure if this is right at all, but this is how I would currently approach it. I just have no idea how to satisfy the constraint given of $a^2<4b$ and that the limit of Yg is = 0.
Any guidance would be greatly appreciated.
ordinary-differential-equations limits
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
$endgroup$
add a comment |
$begingroup$
I was given a theoretical problem in my Linear Algebra and Differentials class and I am stumped on how to approach this problem.
Consider the ODE y''+ay'+by=0 where a and b are constants such that $a^2$ < 4b. What conditions on a and b guarentees that the general solution of the ODE satisfies $displaystyle{lim_{x to infty} Yg=0}$
So I do understand that Yg is the general solution, which contains the homogeneous solution + the particular solution.
So far we have worked with simple polynomials in our class, so what we do is we solve for the particular solution first and use $y=C1e^(r1*x) + C2e^(r2*x)$ where t1 and t2 are the roots to the equation $r^2+ar+b=0$ (from above equation).
The way we solve the particular solution is by "Guessing" the right hand side of this equation, but since it is = 0 I believe that the Yg is just the Y homogeneous.
$Yg = Yh+Yp => Yg = Yh+0 => Yg=Yh$
So in this case I believe my Yg would be of the form $Yg = e^(r1*x) + e^(r2*x)$ which implies that my lim equation is $displaystyle{lim_{x to infty} e^(r1*x) + e^(r2*x)=0}$
I am not sure if this is right at all, but this is how I would currently approach it. I just have no idea how to satisfy the constraint given of $a^2<4b$ and that the limit of Yg is = 0.
Any guidance would be greatly appreciated.
ordinary-differential-equations limits
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
$endgroup$
I was given a theoretical problem in my Linear Algebra and Differentials class and I am stumped on how to approach this problem.
Consider the ODE y''+ay'+by=0 where a and b are constants such that $a^2$ < 4b. What conditions on a and b guarentees that the general solution of the ODE satisfies $displaystyle{lim_{x to infty} Yg=0}$
So I do understand that Yg is the general solution, which contains the homogeneous solution + the particular solution.
So far we have worked with simple polynomials in our class, so what we do is we solve for the particular solution first and use $y=C1e^(r1*x) + C2e^(r2*x)$ where t1 and t2 are the roots to the equation $r^2+ar+b=0$ (from above equation).
The way we solve the particular solution is by "Guessing" the right hand side of this equation, but since it is = 0 I believe that the Yg is just the Y homogeneous.
$Yg = Yh+Yp => Yg = Yh+0 => Yg=Yh$
So in this case I believe my Yg would be of the form $Yg = e^(r1*x) + e^(r2*x)$ which implies that my lim equation is $displaystyle{lim_{x to infty} e^(r1*x) + e^(r2*x)=0}$
I am not sure if this is right at all, but this is how I would currently approach it. I just have no idea how to satisfy the constraint given of $a^2<4b$ and that the limit of Yg is = 0.
Any guidance would be greatly appreciated.
ordinary-differential-equations limits
ordinary-differential-equations limits
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
asked Dec 18 '18 at 23:24
97WaterPolo97WaterPolo
266
266
add a comment |
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