Find the integrating factor and solve
$begingroup$
Find an integrating factor and solve
$(2x^2y + x),dy + (xy^2 + y),dx = 0$
I checked if it was exact, which it wasn't.
Then I found
$M/Y$ to be $xy + 1$
$N/Y$ to be $2xy + 1,$ but when I tried to put it in the form of
$n N/X - mM/Y$ I got $-2xy,$ which didn't really tell me much about the value of $n$ and $m.$
So I tried over by multiplying the original equation by $x^ny^m$ and didn't get two linear equations at the end but two equations, which still had $xy$ terms and am stuck now. Any help would be appreciated.
ordinary-differential-equations
$endgroup$
add a comment |
$begingroup$
Find an integrating factor and solve
$(2x^2y + x),dy + (xy^2 + y),dx = 0$
I checked if it was exact, which it wasn't.
Then I found
$M/Y$ to be $xy + 1$
$N/Y$ to be $2xy + 1,$ but when I tried to put it in the form of
$n N/X - mM/Y$ I got $-2xy,$ which didn't really tell me much about the value of $n$ and $m.$
So I tried over by multiplying the original equation by $x^ny^m$ and didn't get two linear equations at the end but two equations, which still had $xy$ terms and am stuck now. Any help would be appreciated.
ordinary-differential-equations
$endgroup$
add a comment |
$begingroup$
Find an integrating factor and solve
$(2x^2y + x),dy + (xy^2 + y),dx = 0$
I checked if it was exact, which it wasn't.
Then I found
$M/Y$ to be $xy + 1$
$N/Y$ to be $2xy + 1,$ but when I tried to put it in the form of
$n N/X - mM/Y$ I got $-2xy,$ which didn't really tell me much about the value of $n$ and $m.$
So I tried over by multiplying the original equation by $x^ny^m$ and didn't get two linear equations at the end but two equations, which still had $xy$ terms and am stuck now. Any help would be appreciated.
ordinary-differential-equations
$endgroup$
Find an integrating factor and solve
$(2x^2y + x),dy + (xy^2 + y),dx = 0$
I checked if it was exact, which it wasn't.
Then I found
$M/Y$ to be $xy + 1$
$N/Y$ to be $2xy + 1,$ but when I tried to put it in the form of
$n N/X - mM/Y$ I got $-2xy,$ which didn't really tell me much about the value of $n$ and $m.$
So I tried over by multiplying the original equation by $x^ny^m$ and didn't get two linear equations at the end but two equations, which still had $xy$ terms and am stuck now. Any help would be appreciated.
ordinary-differential-equations
ordinary-differential-equations
edited Dec 19 '18 at 18:55
Adrian Keister
5,27371933
5,27371933
asked Dec 19 '18 at 18:48
dragophdragoph
111
111
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
Let's do your $x^ny^m$ and see where we get:
$$underbrace{left(2x^{n+2}y^{m+1}+x^{n+1}y^mright)}_{partial_x},dy+underbrace{left(x^{n+1}y^{m+2}+x^ny^{m+1}right)}_{partial_y},dx=0 $$
begin{align*}
2(n+2)x^{n+1}y^{m+1}+(n+1)x^ny^m&overset{text{set}}{=}(m+2)x^{n+1}y^{m+1}+(m+1)x^ny^m \
2(n+2)xy+n+1&=(m+2)xy+m+1 \
2(n+2)xy+n&=(m+2)xy+m.
end{align*}
So the terms not multiplying $xy$ will have to be equal, forcing $n=m$. Can we get the coefficients of $xy$ to line up? We'd need
begin{align*}
2(n+2)&=n+2\
2n+4&=n+2\
n&=-2=m.
end{align*}
It works! Evidently, the integrating factor is
$$mu(x,y)=frac{1}{x^2y^2}. $$
If you multiply through by this, you'll find the equation is now exact.
$endgroup$
1
$begingroup$
I got the same equation as you of $2(n+2)xy + n = (m+2)xy + m$. I didn't realize you would then set the non multiplying terms and solve. I was expecting I would get something not including xy. Thanks!
$endgroup$
– dragoph
Dec 19 '18 at 19:26
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let's do your $x^ny^m$ and see where we get:
$$underbrace{left(2x^{n+2}y^{m+1}+x^{n+1}y^mright)}_{partial_x},dy+underbrace{left(x^{n+1}y^{m+2}+x^ny^{m+1}right)}_{partial_y},dx=0 $$
begin{align*}
2(n+2)x^{n+1}y^{m+1}+(n+1)x^ny^m&overset{text{set}}{=}(m+2)x^{n+1}y^{m+1}+(m+1)x^ny^m \
2(n+2)xy+n+1&=(m+2)xy+m+1 \
2(n+2)xy+n&=(m+2)xy+m.
end{align*}
So the terms not multiplying $xy$ will have to be equal, forcing $n=m$. Can we get the coefficients of $xy$ to line up? We'd need
begin{align*}
2(n+2)&=n+2\
2n+4&=n+2\
n&=-2=m.
end{align*}
It works! Evidently, the integrating factor is
$$mu(x,y)=frac{1}{x^2y^2}. $$
If you multiply through by this, you'll find the equation is now exact.
$endgroup$
1
$begingroup$
I got the same equation as you of $2(n+2)xy + n = (m+2)xy + m$. I didn't realize you would then set the non multiplying terms and solve. I was expecting I would get something not including xy. Thanks!
$endgroup$
– dragoph
Dec 19 '18 at 19:26
add a comment |
$begingroup$
Let's do your $x^ny^m$ and see where we get:
$$underbrace{left(2x^{n+2}y^{m+1}+x^{n+1}y^mright)}_{partial_x},dy+underbrace{left(x^{n+1}y^{m+2}+x^ny^{m+1}right)}_{partial_y},dx=0 $$
begin{align*}
2(n+2)x^{n+1}y^{m+1}+(n+1)x^ny^m&overset{text{set}}{=}(m+2)x^{n+1}y^{m+1}+(m+1)x^ny^m \
2(n+2)xy+n+1&=(m+2)xy+m+1 \
2(n+2)xy+n&=(m+2)xy+m.
end{align*}
So the terms not multiplying $xy$ will have to be equal, forcing $n=m$. Can we get the coefficients of $xy$ to line up? We'd need
begin{align*}
2(n+2)&=n+2\
2n+4&=n+2\
n&=-2=m.
end{align*}
It works! Evidently, the integrating factor is
$$mu(x,y)=frac{1}{x^2y^2}. $$
If you multiply through by this, you'll find the equation is now exact.
$endgroup$
1
$begingroup$
I got the same equation as you of $2(n+2)xy + n = (m+2)xy + m$. I didn't realize you would then set the non multiplying terms and solve. I was expecting I would get something not including xy. Thanks!
$endgroup$
– dragoph
Dec 19 '18 at 19:26
add a comment |
$begingroup$
Let's do your $x^ny^m$ and see where we get:
$$underbrace{left(2x^{n+2}y^{m+1}+x^{n+1}y^mright)}_{partial_x},dy+underbrace{left(x^{n+1}y^{m+2}+x^ny^{m+1}right)}_{partial_y},dx=0 $$
begin{align*}
2(n+2)x^{n+1}y^{m+1}+(n+1)x^ny^m&overset{text{set}}{=}(m+2)x^{n+1}y^{m+1}+(m+1)x^ny^m \
2(n+2)xy+n+1&=(m+2)xy+m+1 \
2(n+2)xy+n&=(m+2)xy+m.
end{align*}
So the terms not multiplying $xy$ will have to be equal, forcing $n=m$. Can we get the coefficients of $xy$ to line up? We'd need
begin{align*}
2(n+2)&=n+2\
2n+4&=n+2\
n&=-2=m.
end{align*}
It works! Evidently, the integrating factor is
$$mu(x,y)=frac{1}{x^2y^2}. $$
If you multiply through by this, you'll find the equation is now exact.
$endgroup$
Let's do your $x^ny^m$ and see where we get:
$$underbrace{left(2x^{n+2}y^{m+1}+x^{n+1}y^mright)}_{partial_x},dy+underbrace{left(x^{n+1}y^{m+2}+x^ny^{m+1}right)}_{partial_y},dx=0 $$
begin{align*}
2(n+2)x^{n+1}y^{m+1}+(n+1)x^ny^m&overset{text{set}}{=}(m+2)x^{n+1}y^{m+1}+(m+1)x^ny^m \
2(n+2)xy+n+1&=(m+2)xy+m+1 \
2(n+2)xy+n&=(m+2)xy+m.
end{align*}
So the terms not multiplying $xy$ will have to be equal, forcing $n=m$. Can we get the coefficients of $xy$ to line up? We'd need
begin{align*}
2(n+2)&=n+2\
2n+4&=n+2\
n&=-2=m.
end{align*}
It works! Evidently, the integrating factor is
$$mu(x,y)=frac{1}{x^2y^2}. $$
If you multiply through by this, you'll find the equation is now exact.
edited Dec 19 '18 at 19:31
answered Dec 19 '18 at 19:12
Adrian KeisterAdrian Keister
5,27371933
5,27371933
1
$begingroup$
I got the same equation as you of $2(n+2)xy + n = (m+2)xy + m$. I didn't realize you would then set the non multiplying terms and solve. I was expecting I would get something not including xy. Thanks!
$endgroup$
– dragoph
Dec 19 '18 at 19:26
add a comment |
1
$begingroup$
I got the same equation as you of $2(n+2)xy + n = (m+2)xy + m$. I didn't realize you would then set the non multiplying terms and solve. I was expecting I would get something not including xy. Thanks!
$endgroup$
– dragoph
Dec 19 '18 at 19:26
1
1
$begingroup$
I got the same equation as you of $2(n+2)xy + n = (m+2)xy + m$. I didn't realize you would then set the non multiplying terms and solve. I was expecting I would get something not including xy. Thanks!
$endgroup$
– dragoph
Dec 19 '18 at 19:26
$begingroup$
I got the same equation as you of $2(n+2)xy + n = (m+2)xy + m$. I didn't realize you would then set the non multiplying terms and solve. I was expecting I would get something not including xy. Thanks!
$endgroup$
– dragoph
Dec 19 '18 at 19:26
add a comment |
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