Find the general solution to the ODE $xfrac{dy}{dx}=y-frac{1}{y}$
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I have been working through an ODE finding the general solution and following the modulus through the equation has left me with four general solutions, as shown below. Online ODE solvers, however, have only calculated the two (positive and negative), on the right hand side of the page. Am I incorrect with my use of the modulus operator after I integrated each side then?
$$begin{align*}xfrac{dy}{dx}&=y-frac{1}{y}quadquad(-1<y<1)\
intfrac{1}{y-frac{1}{y}} dy&=intfrac{1}{x} dx\
intfrac{y}{y^2-1} dy&=intfrac{1}{x} dx\
frac{1}{2}ln{|y^2-1|}&=ln{|x|}+c\
sqrt{|y^2-1|}&=A|x| text{, where $A=e^c$}\
|y^2-1|&=A^2x^2\
implies y^2-1&=A^2x^2&1-y^2&=A^2x^2\
y^2&=A^2x^2+1&y^2&=1-A^2x^2\
\
therefore y&=pmsqrt{Kx^2+1}&y&=pmsqrt{1-Kx^2} text{, where $K=A^2$}\
end{align*}$$
Thanks :)
calculus integration differential-equations absolute-value
|
show 1 more comment
up vote
0
down vote
favorite
I have been working through an ODE finding the general solution and following the modulus through the equation has left me with four general solutions, as shown below. Online ODE solvers, however, have only calculated the two (positive and negative), on the right hand side of the page. Am I incorrect with my use of the modulus operator after I integrated each side then?
$$begin{align*}xfrac{dy}{dx}&=y-frac{1}{y}quadquad(-1<y<1)\
intfrac{1}{y-frac{1}{y}} dy&=intfrac{1}{x} dx\
intfrac{y}{y^2-1} dy&=intfrac{1}{x} dx\
frac{1}{2}ln{|y^2-1|}&=ln{|x|}+c\
sqrt{|y^2-1|}&=A|x| text{, where $A=e^c$}\
|y^2-1|&=A^2x^2\
implies y^2-1&=A^2x^2&1-y^2&=A^2x^2\
y^2&=A^2x^2+1&y^2&=1-A^2x^2\
\
therefore y&=pmsqrt{Kx^2+1}&y&=pmsqrt{1-Kx^2} text{, where $K=A^2$}\
end{align*}$$
Thanks :)
calculus integration differential-equations absolute-value
1
If $-1<y<1$ then $|y^2-1|=1-y^2$.
– Nosrati
Nov 15 at 12:10
What do you mean by $-1lt ylt -1$
– Digamma
Nov 15 at 12:11
1
$-1 < y < 1$, therefore, $0> y^2 -1$ and you can forget the modulus simply multiplying by $-1$.
– AlgebraicallyClosed
Nov 15 at 12:12
@Nosrati Thanks, got it now. If y could take any value would my method have been correct?
– Adam Bromiley
Nov 15 at 12:19
your method is nice with some typo.
– Nosrati
Nov 15 at 12:19
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have been working through an ODE finding the general solution and following the modulus through the equation has left me with four general solutions, as shown below. Online ODE solvers, however, have only calculated the two (positive and negative), on the right hand side of the page. Am I incorrect with my use of the modulus operator after I integrated each side then?
$$begin{align*}xfrac{dy}{dx}&=y-frac{1}{y}quadquad(-1<y<1)\
intfrac{1}{y-frac{1}{y}} dy&=intfrac{1}{x} dx\
intfrac{y}{y^2-1} dy&=intfrac{1}{x} dx\
frac{1}{2}ln{|y^2-1|}&=ln{|x|}+c\
sqrt{|y^2-1|}&=A|x| text{, where $A=e^c$}\
|y^2-1|&=A^2x^2\
implies y^2-1&=A^2x^2&1-y^2&=A^2x^2\
y^2&=A^2x^2+1&y^2&=1-A^2x^2\
\
therefore y&=pmsqrt{Kx^2+1}&y&=pmsqrt{1-Kx^2} text{, where $K=A^2$}\
end{align*}$$
Thanks :)
calculus integration differential-equations absolute-value
I have been working through an ODE finding the general solution and following the modulus through the equation has left me with four general solutions, as shown below. Online ODE solvers, however, have only calculated the two (positive and negative), on the right hand side of the page. Am I incorrect with my use of the modulus operator after I integrated each side then?
$$begin{align*}xfrac{dy}{dx}&=y-frac{1}{y}quadquad(-1<y<1)\
intfrac{1}{y-frac{1}{y}} dy&=intfrac{1}{x} dx\
intfrac{y}{y^2-1} dy&=intfrac{1}{x} dx\
frac{1}{2}ln{|y^2-1|}&=ln{|x|}+c\
sqrt{|y^2-1|}&=A|x| text{, where $A=e^c$}\
|y^2-1|&=A^2x^2\
implies y^2-1&=A^2x^2&1-y^2&=A^2x^2\
y^2&=A^2x^2+1&y^2&=1-A^2x^2\
\
therefore y&=pmsqrt{Kx^2+1}&y&=pmsqrt{1-Kx^2} text{, where $K=A^2$}\
end{align*}$$
Thanks :)
calculus integration differential-equations absolute-value
calculus integration differential-equations absolute-value
edited Nov 15 at 14:26
LutzL
53.6k41953
53.6k41953
asked Nov 15 at 12:05
Adam Bromiley
26218
26218
1
If $-1<y<1$ then $|y^2-1|=1-y^2$.
– Nosrati
Nov 15 at 12:10
What do you mean by $-1lt ylt -1$
– Digamma
Nov 15 at 12:11
1
$-1 < y < 1$, therefore, $0> y^2 -1$ and you can forget the modulus simply multiplying by $-1$.
– AlgebraicallyClosed
Nov 15 at 12:12
@Nosrati Thanks, got it now. If y could take any value would my method have been correct?
– Adam Bromiley
Nov 15 at 12:19
your method is nice with some typo.
– Nosrati
Nov 15 at 12:19
|
show 1 more comment
1
If $-1<y<1$ then $|y^2-1|=1-y^2$.
– Nosrati
Nov 15 at 12:10
What do you mean by $-1lt ylt -1$
– Digamma
Nov 15 at 12:11
1
$-1 < y < 1$, therefore, $0> y^2 -1$ and you can forget the modulus simply multiplying by $-1$.
– AlgebraicallyClosed
Nov 15 at 12:12
@Nosrati Thanks, got it now. If y could take any value would my method have been correct?
– Adam Bromiley
Nov 15 at 12:19
your method is nice with some typo.
– Nosrati
Nov 15 at 12:19
1
1
If $-1<y<1$ then $|y^2-1|=1-y^2$.
– Nosrati
Nov 15 at 12:10
If $-1<y<1$ then $|y^2-1|=1-y^2$.
– Nosrati
Nov 15 at 12:10
What do you mean by $-1lt ylt -1$
– Digamma
Nov 15 at 12:11
What do you mean by $-1lt ylt -1$
– Digamma
Nov 15 at 12:11
1
1
$-1 < y < 1$, therefore, $0> y^2 -1$ and you can forget the modulus simply multiplying by $-1$.
– AlgebraicallyClosed
Nov 15 at 12:12
$-1 < y < 1$, therefore, $0> y^2 -1$ and you can forget the modulus simply multiplying by $-1$.
– AlgebraicallyClosed
Nov 15 at 12:12
@Nosrati Thanks, got it now. If y could take any value would my method have been correct?
– Adam Bromiley
Nov 15 at 12:19
@Nosrati Thanks, got it now. If y could take any value would my method have been correct?
– Adam Bromiley
Nov 15 at 12:19
your method is nice with some typo.
– Nosrati
Nov 15 at 12:19
your method is nice with some typo.
– Nosrati
Nov 15 at 12:19
|
show 1 more comment
3 Answers
3
active
oldest
votes
up vote
1
down vote
accepted
If $-1<y<1$ then $|y^2-1|=1-y^2$.
Another approach: Write the equation as
$$dfrac{x dy-y dx}{x^2}=dfrac{-1}{x^2y} dx$$
or
$$left(dfrac{y}{x}right)dleft(dfrac{y}{x}right)=dfrac{-1}{x^3} dx$$
and
$$left(dfrac{y}{x}right)^2=dfrac{1}{x^2}+C$$
Does this answer the question ?
– Yves Daoust
Nov 15 at 16:20
I should have addressed this comment to the OP. He was worried by his four solutions and usage of the modulus...
– Yves Daoust
Nov 15 at 16:23
So here you are :)
– Nosrati
Nov 15 at 16:24
add a comment |
up vote
0
down vote
Another approach is this. Rewrite the DE as
$$
frac{yy'}{y^2-1}=frac 1x
$$
Which is the same as
$$
frac 12 frac{d}{dx}ln(y^2-1)=frac 1x
$$
Upon integration, one gets
$$
lndfrac{y^2(x)-1}{y^2(x_0)-1}=2lnfrac{x}{x_0}
$$
Hence, one obtains
$$
y^2(x)=1+[y^2(x_0)-1]left(frac{x}{x_0}right)^2
$$
add a comment |
up vote
0
down vote
Your solutions are $pmsqrt{1pm Kx^2}$, where $K$ is positive. But this does not differ from $pmsqrt{1+Kx^2}$ where $K$ is unconstrained.
From
$$frac12log|y^2-1|=log|x|+C$$ you draw
$$log|y^2-1|=log(e^Cx^2)$$ and
$$|y^2-1|=e^Cx.$$
Then
$$y^2=1pm e^Cx^2$$ which you can very well rewrite as
$$y^2=1+Dx^2.$$
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
If $-1<y<1$ then $|y^2-1|=1-y^2$.
Another approach: Write the equation as
$$dfrac{x dy-y dx}{x^2}=dfrac{-1}{x^2y} dx$$
or
$$left(dfrac{y}{x}right)dleft(dfrac{y}{x}right)=dfrac{-1}{x^3} dx$$
and
$$left(dfrac{y}{x}right)^2=dfrac{1}{x^2}+C$$
Does this answer the question ?
– Yves Daoust
Nov 15 at 16:20
I should have addressed this comment to the OP. He was worried by his four solutions and usage of the modulus...
– Yves Daoust
Nov 15 at 16:23
So here you are :)
– Nosrati
Nov 15 at 16:24
add a comment |
up vote
1
down vote
accepted
If $-1<y<1$ then $|y^2-1|=1-y^2$.
Another approach: Write the equation as
$$dfrac{x dy-y dx}{x^2}=dfrac{-1}{x^2y} dx$$
or
$$left(dfrac{y}{x}right)dleft(dfrac{y}{x}right)=dfrac{-1}{x^3} dx$$
and
$$left(dfrac{y}{x}right)^2=dfrac{1}{x^2}+C$$
Does this answer the question ?
– Yves Daoust
Nov 15 at 16:20
I should have addressed this comment to the OP. He was worried by his four solutions and usage of the modulus...
– Yves Daoust
Nov 15 at 16:23
So here you are :)
– Nosrati
Nov 15 at 16:24
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
If $-1<y<1$ then $|y^2-1|=1-y^2$.
Another approach: Write the equation as
$$dfrac{x dy-y dx}{x^2}=dfrac{-1}{x^2y} dx$$
or
$$left(dfrac{y}{x}right)dleft(dfrac{y}{x}right)=dfrac{-1}{x^3} dx$$
and
$$left(dfrac{y}{x}right)^2=dfrac{1}{x^2}+C$$
If $-1<y<1$ then $|y^2-1|=1-y^2$.
Another approach: Write the equation as
$$dfrac{x dy-y dx}{x^2}=dfrac{-1}{x^2y} dx$$
or
$$left(dfrac{y}{x}right)dleft(dfrac{y}{x}right)=dfrac{-1}{x^3} dx$$
and
$$left(dfrac{y}{x}right)^2=dfrac{1}{x^2}+C$$
answered Nov 15 at 12:19
Nosrati
26k62352
26k62352
Does this answer the question ?
– Yves Daoust
Nov 15 at 16:20
I should have addressed this comment to the OP. He was worried by his four solutions and usage of the modulus...
– Yves Daoust
Nov 15 at 16:23
So here you are :)
– Nosrati
Nov 15 at 16:24
add a comment |
Does this answer the question ?
– Yves Daoust
Nov 15 at 16:20
I should have addressed this comment to the OP. He was worried by his four solutions and usage of the modulus...
– Yves Daoust
Nov 15 at 16:23
So here you are :)
– Nosrati
Nov 15 at 16:24
Does this answer the question ?
– Yves Daoust
Nov 15 at 16:20
Does this answer the question ?
– Yves Daoust
Nov 15 at 16:20
I should have addressed this comment to the OP. He was worried by his four solutions and usage of the modulus...
– Yves Daoust
Nov 15 at 16:23
I should have addressed this comment to the OP. He was worried by his four solutions and usage of the modulus...
– Yves Daoust
Nov 15 at 16:23
So here you are :)
– Nosrati
Nov 15 at 16:24
So here you are :)
– Nosrati
Nov 15 at 16:24
add a comment |
up vote
0
down vote
Another approach is this. Rewrite the DE as
$$
frac{yy'}{y^2-1}=frac 1x
$$
Which is the same as
$$
frac 12 frac{d}{dx}ln(y^2-1)=frac 1x
$$
Upon integration, one gets
$$
lndfrac{y^2(x)-1}{y^2(x_0)-1}=2lnfrac{x}{x_0}
$$
Hence, one obtains
$$
y^2(x)=1+[y^2(x_0)-1]left(frac{x}{x_0}right)^2
$$
add a comment |
up vote
0
down vote
Another approach is this. Rewrite the DE as
$$
frac{yy'}{y^2-1}=frac 1x
$$
Which is the same as
$$
frac 12 frac{d}{dx}ln(y^2-1)=frac 1x
$$
Upon integration, one gets
$$
lndfrac{y^2(x)-1}{y^2(x_0)-1}=2lnfrac{x}{x_0}
$$
Hence, one obtains
$$
y^2(x)=1+[y^2(x_0)-1]left(frac{x}{x_0}right)^2
$$
add a comment |
up vote
0
down vote
up vote
0
down vote
Another approach is this. Rewrite the DE as
$$
frac{yy'}{y^2-1}=frac 1x
$$
Which is the same as
$$
frac 12 frac{d}{dx}ln(y^2-1)=frac 1x
$$
Upon integration, one gets
$$
lndfrac{y^2(x)-1}{y^2(x_0)-1}=2lnfrac{x}{x_0}
$$
Hence, one obtains
$$
y^2(x)=1+[y^2(x_0)-1]left(frac{x}{x_0}right)^2
$$
Another approach is this. Rewrite the DE as
$$
frac{yy'}{y^2-1}=frac 1x
$$
Which is the same as
$$
frac 12 frac{d}{dx}ln(y^2-1)=frac 1x
$$
Upon integration, one gets
$$
lndfrac{y^2(x)-1}{y^2(x_0)-1}=2lnfrac{x}{x_0}
$$
Hence, one obtains
$$
y^2(x)=1+[y^2(x_0)-1]left(frac{x}{x_0}right)^2
$$
answered Nov 15 at 16:13
minimax
48518
48518
add a comment |
add a comment |
up vote
0
down vote
Your solutions are $pmsqrt{1pm Kx^2}$, where $K$ is positive. But this does not differ from $pmsqrt{1+Kx^2}$ where $K$ is unconstrained.
From
$$frac12log|y^2-1|=log|x|+C$$ you draw
$$log|y^2-1|=log(e^Cx^2)$$ and
$$|y^2-1|=e^Cx.$$
Then
$$y^2=1pm e^Cx^2$$ which you can very well rewrite as
$$y^2=1+Dx^2.$$
add a comment |
up vote
0
down vote
Your solutions are $pmsqrt{1pm Kx^2}$, where $K$ is positive. But this does not differ from $pmsqrt{1+Kx^2}$ where $K$ is unconstrained.
From
$$frac12log|y^2-1|=log|x|+C$$ you draw
$$log|y^2-1|=log(e^Cx^2)$$ and
$$|y^2-1|=e^Cx.$$
Then
$$y^2=1pm e^Cx^2$$ which you can very well rewrite as
$$y^2=1+Dx^2.$$
add a comment |
up vote
0
down vote
up vote
0
down vote
Your solutions are $pmsqrt{1pm Kx^2}$, where $K$ is positive. But this does not differ from $pmsqrt{1+Kx^2}$ where $K$ is unconstrained.
From
$$frac12log|y^2-1|=log|x|+C$$ you draw
$$log|y^2-1|=log(e^Cx^2)$$ and
$$|y^2-1|=e^Cx.$$
Then
$$y^2=1pm e^Cx^2$$ which you can very well rewrite as
$$y^2=1+Dx^2.$$
Your solutions are $pmsqrt{1pm Kx^2}$, where $K$ is positive. But this does not differ from $pmsqrt{1+Kx^2}$ where $K$ is unconstrained.
From
$$frac12log|y^2-1|=log|x|+C$$ you draw
$$log|y^2-1|=log(e^Cx^2)$$ and
$$|y^2-1|=e^Cx.$$
Then
$$y^2=1pm e^Cx^2$$ which you can very well rewrite as
$$y^2=1+Dx^2.$$
edited Nov 15 at 16:31
answered Nov 15 at 16:21
Yves Daoust
121k668216
121k668216
add a comment |
add a comment |
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1
If $-1<y<1$ then $|y^2-1|=1-y^2$.
– Nosrati
Nov 15 at 12:10
What do you mean by $-1lt ylt -1$
– Digamma
Nov 15 at 12:11
1
$-1 < y < 1$, therefore, $0> y^2 -1$ and you can forget the modulus simply multiplying by $-1$.
– AlgebraicallyClosed
Nov 15 at 12:12
@Nosrati Thanks, got it now. If y could take any value would my method have been correct?
– Adam Bromiley
Nov 15 at 12:19
your method is nice with some typo.
– Nosrati
Nov 15 at 12:19