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$lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone. True/false?
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Let $f: [0,infty) rightarrow mathbbR $ be a continuous function. Consider the ordinary differential equation
$$y'(x) = f(x) y(x) ,quad x> 0 ,quad y(0) = y_0 neq 0.$$
Which of the following statements are true?
$1$. If $int_0^infty|f(x)|dx < infty$, then $y$ is bounded.
$2$. If $int_0^infty|f(x)|dx < infty$, then $lim_x to infty y(x) $ exists.
$3$. If $lim_x to infty f(x) = 1 $, then $lim_x to infty |y(x)| = infty$.
$4$. If $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone.
My attempts: I know that options $1$, $2$, and $3$ are correct...but I'm doubting that option $4$ is true.
Please help me; any hints/solutions will be appreciated. Thank you.
differential-equations
edited Jul 20 at 3:07
anomaly
16.4k42561
asked Jul 19 at 16:04
stupid
55819
add a comment |Â
up vote
0
down vote
favorite
Let $f: [0,infty) rightarrow mathbbR $ be a continuous function. Consider the ordinary differential equation
$$y'(x) = f(x) y(x) ,quad x> 0 ,quad y(0) = y_0 neq 0.$$
Which of the following statements are true?
$1$. If $int_0^infty|f(x)|dx < infty$, then $y$ is bounded.
$2$. If $int_0^infty|f(x)|dx < infty$, then $lim_x to infty y(x) $ exists.
$3$. If $lim_x to infty f(x) = 1 $, then $lim_x to infty |y(x)| = infty$.
$4$. If $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone.
My attempts: I know that options $1$, $2$, and $3$ are correct...but I'm doubting that option $4$ is true.
Please help me; any hints/solutions will be appreciated. Thank you.
differential-equations
edited Jul 20 at 3:07
anomaly
16.4k42561
asked Jul 19 at 16:04
stupid
55819
3
You can write an explicit solution to the ODE. What is the derivative of $ln f$?
– copper.hat
Jul 19 at 16:11
@ copper .hat that mean option 4 is True,,,if $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone..can u elaborate more
– stupid
Jul 19 at 16:12
1
Well, $y$ is monotone iff $y'$ does not change sign. What can you say about the sign of $y$ given that you have a formula for $y$?
– copper.hat
Jul 19 at 16:27
okk that mean y will not monotone..am I right ??
– stupid
Jul 19 at 16:29
1
Well, the sign of $y$ will not change, but you can change $f$ so that $y'$ changes sign and still have $f(x) to 1$.
– copper.hat
Jul 19 at 16:33
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f: [0,infty) rightarrow mathbbR $ be a continuous function. Consider the ordinary differential equation
$$y'(x) = f(x) y(x) ,quad x> 0 ,quad y(0) = y_0 neq 0.$$
Which of the following statements are true?
$1$. If $int_0^infty|f(x)|dx < infty$, then $y$ is bounded.
$2$. If $int_0^infty|f(x)|dx < infty$, then $lim_x to infty y(x) $ exists.
$3$. If $lim_x to infty f(x) = 1 $, then $lim_x to infty |y(x)| = infty$.
$4$. If $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone.
My attempts: I know that options $1$, $2$, and $3$ are correct...but I'm doubting that option $4$ is true.
Please help me; any hints/solutions will be appreciated. Thank you.
differential-equations
edited Jul 20 at 3:07
anomaly
16.4k42561
asked Jul 19 at 16:04
stupid
55819
Let $f: [0,infty) rightarrow mathbbR $ be a continuous function. Consider the ordinary differential equation
$$y'(x) = f(x) y(x) ,quad x> 0 ,quad y(0) = y_0 neq 0.$$
Which of the following statements are true?
$1$. If $int_0^infty|f(x)|dx < infty$, then $y$ is bounded.
$2$. If $int_0^infty|f(x)|dx < infty$, then $lim_x to infty y(x) $ exists.
$3$. If $lim_x to infty f(x) = 1 $, then $lim_x to infty |y(x)| = infty$.
$4$. If $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone.
My attempts: I know that options $1$, $2$, and $3$ are correct...but I'm doubting that option $4$ is true.
Please help me; any hints/solutions will be appreciated. Thank you.
differential-equations
edited Jul 20 at 3:07
anomaly
16.4k42561
asked Jul 19 at 16:04
stupid
55819
edited Jul 20 at 3:07
anomaly
16.4k42561
edited Jul 20 at 3:07
anomaly
16.4k42561
edited Jul 20 at 3:07
anomaly
16.4k42561
16.4k42561
asked Jul 19 at 16:04
stupid
55819
asked Jul 19 at 16:04
stupid
55819
asked Jul 19 at 16:04
stupid
55819
55819
3
You can write an explicit solution to the ODE. What is the derivative of $ln f$?
– copper.hat
Jul 19 at 16:11
@ copper .hat that mean option 4 is True,,,if $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone..can u elaborate more
– stupid
Jul 19 at 16:12
1
Well, $y$ is monotone iff $y'$ does not change sign. What can you say about the sign of $y$ given that you have a formula for $y$?
– copper.hat
Jul 19 at 16:27
okk that mean y will not monotone..am I right ??
– stupid
Jul 19 at 16:29
1
Well, the sign of $y$ will not change, but you can change $f$ so that $y'$ changes sign and still have $f(x) to 1$.
– copper.hat
Jul 19 at 16:33
add a comment |Â
3
You can write an explicit solution to the ODE. What is the derivative of $ln f$?
– copper.hat
Jul 19 at 16:11
@ copper .hat that mean option 4 is True,,,if $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone..can u elaborate more
– stupid
Jul 19 at 16:12
1
Well, $y$ is monotone iff $y'$ does not change sign. What can you say about the sign of $y$ given that you have a formula for $y$?
– copper.hat
Jul 19 at 16:27
okk that mean y will not monotone..am I right ??
– stupid
Jul 19 at 16:29
1
Well, the sign of $y$ will not change, but you can change $f$ so that $y'$ changes sign and still have $f(x) to 1$.
– copper.hat
Jul 19 at 16:33
3
3
You can write an explicit solution to the ODE. What is the derivative of $ln f$?
– copper.hat
Jul 19 at 16:11
You can write an explicit solution to the ODE. What is the derivative of $ln f$?
– copper.hat
Jul 19 at 16:11
@ copper .hat that mean option 4 is True,,,if $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone..can u elaborate more
– stupid
Jul 19 at 16:12
@ copper .hat that mean option 4 is True,,,if $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone..can u elaborate more
– stupid
Jul 19 at 16:12
1
1
Well, $y$ is monotone iff $y'$ does not change sign. What can you say about the sign of $y$ given that you have a formula for $y$?
– copper.hat
Jul 19 at 16:27
Well, $y$ is monotone iff $y'$ does not change sign. What can you say about the sign of $y$ given that you have a formula for $y$?
– copper.hat
Jul 19 at 16:27
okk that mean y will not monotone..am I right ??
– stupid
Jul 19 at 16:29
okk that mean y will not monotone..am I right ??
– stupid
Jul 19 at 16:29
1
1
Well, the sign of $y$ will not change, but you can change $f$ so that $y'$ changes sign and still have $f(x) to 1$.
– copper.hat
Jul 19 at 16:33
Well, the sign of $y$ will not change, but you can change $f$ so that $y'$ changes sign and still have $f(x) to 1$.
– copper.hat
Jul 19 at 16:33
add a comment |Â
1 Answer
1
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up vote
1
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Suppose that $f$ is continuous and $y_0>0$ and $y(x)=y_0cdot e^g(x)$ where $g(x)=int_0^xf(u)du.$ Suppose $f(1)=f(2)=0$ and that $f(x)>0$ for $xin [0,1)cup (2,infty)$ and $f(x)<0$ for $xin (1,2).$ And suppose that $lim_xto inftyf(x)=1.$
We have $y(2)-y(1)=$ $y_0cdot e^g(1)cdot (e^g(2)-g(1)-1).$ Now $g(2)-g(1)=int_1^2f(u)du<0$ so $f(2)<f(1).$
We have $y(1)-y(0)=y_0cdot e^g(0)cdot (e^g(1)-g(0)-1).$ Now $g(1)-g(0)=int_0^1 f(u)du>0$ so $f(1)>f(0).$
Remark. If $y>0$ then $yf=y'implies f=y'/y=(ln y)'$ which implies that $ln y$ is an anti-derivative of $f$.
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Suppose that $f$ is continuous and $y_0>0$ and $y(x)=y_0cdot e^g(x)$ where $g(x)=int_0^xf(u)du.$ Suppose $f(1)=f(2)=0$ and that $f(x)>0$ for $xin [0,1)cup (2,infty)$ and $f(x)<0$ for $xin (1,2).$ And suppose that $lim_xto inftyf(x)=1.$
We have $y(2)-y(1)=$ $y_0cdot e^g(1)cdot (e^g(2)-g(1)-1).$ Now $g(2)-g(1)=int_1^2f(u)du<0$ so $f(2)<f(1).$
We have $y(1)-y(0)=y_0cdot e^g(0)cdot (e^g(1)-g(0)-1).$ Now $g(1)-g(0)=int_0^1 f(u)du>0$ so $f(1)>f(0).$
Remark. If $y>0$ then $yf=y'implies f=y'/y=(ln y)'$ which implies that $ln y$ is an anti-derivative of $f$.
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
add a comment |Â
up vote
1
down vote
accepted
Suppose that $f$ is continuous and $y_0>0$ and $y(x)=y_0cdot e^g(x)$ where $g(x)=int_0^xf(u)du.$ Suppose $f(1)=f(2)=0$ and that $f(x)>0$ for $xin [0,1)cup (2,infty)$ and $f(x)<0$ for $xin (1,2).$ And suppose that $lim_xto inftyf(x)=1.$
We have $y(2)-y(1)=$ $y_0cdot e^g(1)cdot (e^g(2)-g(1)-1).$ Now $g(2)-g(1)=int_1^2f(u)du<0$ so $f(2)<f(1).$
We have $y(1)-y(0)=y_0cdot e^g(0)cdot (e^g(1)-g(0)-1).$ Now $g(1)-g(0)=int_0^1 f(u)du>0$ so $f(1)>f(0).$
Remark. If $y>0$ then $yf=y'implies f=y'/y=(ln y)'$ which implies that $ln y$ is an anti-derivative of $f$.
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Suppose that $f$ is continuous and $y_0>0$ and $y(x)=y_0cdot e^g(x)$ where $g(x)=int_0^xf(u)du.$ Suppose $f(1)=f(2)=0$ and that $f(x)>0$ for $xin [0,1)cup (2,infty)$ and $f(x)<0$ for $xin (1,2).$ And suppose that $lim_xto inftyf(x)=1.$
We have $y(2)-y(1)=$ $y_0cdot e^g(1)cdot (e^g(2)-g(1)-1).$ Now $g(2)-g(1)=int_1^2f(u)du<0$ so $f(2)<f(1).$
We have $y(1)-y(0)=y_0cdot e^g(0)cdot (e^g(1)-g(0)-1).$ Now $g(1)-g(0)=int_0^1 f(u)du>0$ so $f(1)>f(0).$
Remark. If $y>0$ then $yf=y'implies f=y'/y=(ln y)'$ which implies that $ln y$ is an anti-derivative of $f$.
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
Suppose that $f$ is continuous and $y_0>0$ and $y(x)=y_0cdot e^g(x)$ where $g(x)=int_0^xf(u)du.$ Suppose $f(1)=f(2)=0$ and that $f(x)>0$ for $xin [0,1)cup (2,infty)$ and $f(x)<0$ for $xin (1,2).$ And suppose that $lim_xto inftyf(x)=1.$
We have $y(2)-y(1)=$ $y_0cdot e^g(1)cdot (e^g(2)-g(1)-1).$ Now $g(2)-g(1)=int_1^2f(u)du<0$ so $f(2)<f(1).$
We have $y(1)-y(0)=y_0cdot e^g(0)cdot (e^g(1)-g(0)-1).$ Now $g(1)-g(0)=int_0^1 f(u)du>0$ so $f(1)>f(0).$
Remark. If $y>0$ then $yf=y'implies f=y'/y=(ln y)'$ which implies that $ln y$ is an anti-derivative of $f$.
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
edited Jul 20 at 7:25
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
answered Jul 20 at 2:41
DanielWainfleet
31.7k31643
31.7k31643
add a comment |Â
add a comment |Â
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3
You can write an explicit solution to the ODE. What is the derivative of $ln f$?
– copper.hat
Jul 19 at 16:11
@ copper .hat that mean option 4 is True,,,if $lim_x rightarrow infty f(x) = 1 $ then $y $ is monotone..can u elaborate more
– stupid
Jul 19 at 16:12
1
Well, $y$ is monotone iff $y'$ does not change sign. What can you say about the sign of $y$ given that you have a formula for $y$?
– copper.hat
Jul 19 at 16:27
okk that mean y will not monotone..am I right ??
– stupid
Jul 19 at 16:29
1
Well, the sign of $y$ will not change, but you can change $f$ so that $y'$ changes sign and still have $f(x) to 1$.
– copper.hat
Jul 19 at 16:33