Solving the linear first order differential equation?
Clash Royale CLAN TAG#URR8PPP
$begingroup$
I was given the equation
$y' = -xy$, where $y(0) = 1$.
My solution was as follows:
$$fracdydx = -xy $$
$$dy = -xy dx $$
$$int dy = int -xy dx $$
$$y = -fracx^22y + c $$
$$y + fracx^22y = c $$
$$y(1 + fracx^22) = c $$
$$y = fracc1 + fracx^22 $$
I tried plugging in $0$ and $1$ respectively to find $c$, and got $c = 0$.
I just know that what I did is wrong. Where did I go wrong, please? Thanks!
integration ordinary-differential-equations derivatives
$endgroup$
add a comment |
$begingroup$
I was given the equation
$y' = -xy$, where $y(0) = 1$.
My solution was as follows:
$$fracdydx = -xy $$
$$dy = -xy dx $$
$$int dy = int -xy dx $$
$$y = -fracx^22y + c $$
$$y + fracx^22y = c $$
$$y(1 + fracx^22) = c $$
$$y = fracc1 + fracx^22 $$
I tried plugging in $0$ and $1$ respectively to find $c$, and got $c = 0$.
I just know that what I did is wrong. Where did I go wrong, please? Thanks!
integration ordinary-differential-equations derivatives
$endgroup$
1
$begingroup$
Why don't you separate the variables like $$fracdyy=-xdx?$$ Also the final conclusion is definitely wrong.
$endgroup$
– Fakemistake
Feb 24 at 9:20
add a comment |
$begingroup$
I was given the equation
$y' = -xy$, where $y(0) = 1$.
My solution was as follows:
$$fracdydx = -xy $$
$$dy = -xy dx $$
$$int dy = int -xy dx $$
$$y = -fracx^22y + c $$
$$y + fracx^22y = c $$
$$y(1 + fracx^22) = c $$
$$y = fracc1 + fracx^22 $$
I tried plugging in $0$ and $1$ respectively to find $c$, and got $c = 0$.
I just know that what I did is wrong. Where did I go wrong, please? Thanks!
integration ordinary-differential-equations derivatives
$endgroup$
I was given the equation
$y' = -xy$, where $y(0) = 1$.
My solution was as follows:
$$fracdydx = -xy $$
$$dy = -xy dx $$
$$int dy = int -xy dx $$
$$y = -fracx^22y + c $$
$$y + fracx^22y = c $$
$$y(1 + fracx^22) = c $$
$$y = fracc1 + fracx^22 $$
I tried plugging in $0$ and $1$ respectively to find $c$, and got $c = 0$.
I just know that what I did is wrong. Where did I go wrong, please? Thanks!
integration ordinary-differential-equations derivatives
integration ordinary-differential-equations derivatives
edited Mar 15 at 4:55
Paras Khosla
2,348222
2,348222
asked Feb 24 at 8:53
A.SmithA.Smith
262
262
1
$begingroup$
Why don't you separate the variables like $$fracdyy=-xdx?$$ Also the final conclusion is definitely wrong.
$endgroup$
– Fakemistake
Feb 24 at 9:20
add a comment |
1
$begingroup$
Why don't you separate the variables like $$fracdyy=-xdx?$$ Also the final conclusion is definitely wrong.
$endgroup$
– Fakemistake
Feb 24 at 9:20
1
1
$begingroup$
Why don't you separate the variables like $$fracdyy=-xdx?$$ Also the final conclusion is definitely wrong.
$endgroup$
– Fakemistake
Feb 24 at 9:20
$begingroup$
Why don't you separate the variables like $$fracdyy=-xdx?$$ Also the final conclusion is definitely wrong.
$endgroup$
– Fakemistake
Feb 24 at 9:20
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
Hint:
This is a variable separable differential equation that is we can separate the differential equation such that for some functions $f(x)$ and $g(y)$: $f(x)mathrm dx =g(y)mathrm dy$. $$dfracmathrm dymathrm dx=-xy implies int dfrac1ymathrm dy =-int x mathrm dx$$
Now all that is left is to compute the integral and use the inital condition to figure out the value of constant. Can you proceed?
Note that you cannot simply integrate $-xy$ wrt $x$. Your computation assumes that $y$ is a constant which it most certainly is not.
$endgroup$
1
$begingroup$
That helped a lot! Thank you!
$endgroup$
– A.Smith
Feb 24 at 15:25
add a comment |
$begingroup$
(Too long for a comment.)
No one else has mentioned this, so I will...
Remember: $y$ is a function of $x$, not something that is constant with respect to variation of $x$. (Forgetting this is a fairly common error when first performing implicit differentiation.) It can help to write "$y(x)$" rather than just "$y$".
Your error is believing $int -x y(x) , mathrmdx = frac-12x^2 y$. But $y$ is not some constant in this integral; it is a function that depends on $x$. Otherwise, what is written could be the absurdity (when, say $y(x) = frac1x^2$), "$int -x cdot frac1x^2 ,mathrmdx = frac-12 x^2 cdot frac1x^2 + C_1 = C$" rather than the correct
$$ int -x cdot frac1x^2 ,mathrmdx = ln|x| + C text. $$
Quick way to verify this: Implicitly differentiate your antiderivative to see if you get the integrand and differential element back:beginalign*
left( frac-12x^2 y(x) right)'
&= frac-12x^2 (y(x))' + left( frac-12x^2 right)' y(x) \
&= frac-12x^2 ,mathrmdy(x) + left( frac-12 cdot 2 x ,mathrmdx right) y(x) \
&= frac-12x^2 ,mathrmdy(x) - x y(x) ,mathrmdx text,
endalign*
which isn't quite "$- x y,mathrmdx$".
(However, we have shown beginalign*
int left( frac-12x^2 fracmathrmdy(x)mathrmdx - x y(x) right) ,mathrmdx
&= int fracmathrmdmathrmdx left( frac-12x^2 y(x) right) , mathrmd x \
&= frac-12x^2 y(x) + C text.
endalign*
Familiarity with this kind of manipulation could be useful in the future.)
$endgroup$
add a comment |
$begingroup$
After writing
$$intfracmathrmdyy=int -x,mathrmdx,$$
you get $ln|y|=-fracx^22+c_1$, which implies
$$y(x)=cexp(-x^2/2)quadtextforquad c=pmexp(c_1).$$
After plugging in $y(0)=1$, you get $c=1$, so
$$y(x)=exp(-x^2/2).$$
$endgroup$
$begingroup$
Thank you so so much! I super appreciate it!
$endgroup$
– A.Smith
Feb 24 at 15:53
$begingroup$
You're welcome :)
$endgroup$
– st.math
Feb 24 at 16:32
add a comment |
$begingroup$
Here is another method that does not use separation of variables and uses integrating factors. Write
$$
y'+xy=0
$$
and multiply both sides by $e^int x, dx=e^x^2/2$ to get that
$$
0=y'e^x^2/2+xye^x^2/2=fracddx(ye^x^2/2).
$$
So
$$
ye^x^2/2=cimplies y=ce^-x^2/2
$$
for some $c$. Since $y(0)=1$, we deduce that
$$
y=exp(-x^2/2).
$$
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
This is a variable separable differential equation that is we can separate the differential equation such that for some functions $f(x)$ and $g(y)$: $f(x)mathrm dx =g(y)mathrm dy$. $$dfracmathrm dymathrm dx=-xy implies int dfrac1ymathrm dy =-int x mathrm dx$$
Now all that is left is to compute the integral and use the inital condition to figure out the value of constant. Can you proceed?
Note that you cannot simply integrate $-xy$ wrt $x$. Your computation assumes that $y$ is a constant which it most certainly is not.
$endgroup$
1
$begingroup$
That helped a lot! Thank you!
$endgroup$
– A.Smith
Feb 24 at 15:25
add a comment |
$begingroup$
Hint:
This is a variable separable differential equation that is we can separate the differential equation such that for some functions $f(x)$ and $g(y)$: $f(x)mathrm dx =g(y)mathrm dy$. $$dfracmathrm dymathrm dx=-xy implies int dfrac1ymathrm dy =-int x mathrm dx$$
Now all that is left is to compute the integral and use the inital condition to figure out the value of constant. Can you proceed?
Note that you cannot simply integrate $-xy$ wrt $x$. Your computation assumes that $y$ is a constant which it most certainly is not.
$endgroup$
1
$begingroup$
That helped a lot! Thank you!
$endgroup$
– A.Smith
Feb 24 at 15:25
add a comment |
$begingroup$
Hint:
This is a variable separable differential equation that is we can separate the differential equation such that for some functions $f(x)$ and $g(y)$: $f(x)mathrm dx =g(y)mathrm dy$. $$dfracmathrm dymathrm dx=-xy implies int dfrac1ymathrm dy =-int x mathrm dx$$
Now all that is left is to compute the integral and use the inital condition to figure out the value of constant. Can you proceed?
Note that you cannot simply integrate $-xy$ wrt $x$. Your computation assumes that $y$ is a constant which it most certainly is not.
$endgroup$
Hint:
This is a variable separable differential equation that is we can separate the differential equation such that for some functions $f(x)$ and $g(y)$: $f(x)mathrm dx =g(y)mathrm dy$. $$dfracmathrm dymathrm dx=-xy implies int dfrac1ymathrm dy =-int x mathrm dx$$
Now all that is left is to compute the integral and use the inital condition to figure out the value of constant. Can you proceed?
Note that you cannot simply integrate $-xy$ wrt $x$. Your computation assumes that $y$ is a constant which it most certainly is not.
edited Feb 24 at 9:10
answered Feb 24 at 8:55
Paras KhoslaParas Khosla
2,348222
2,348222
1
$begingroup$
That helped a lot! Thank you!
$endgroup$
– A.Smith
Feb 24 at 15:25
add a comment |
1
$begingroup$
That helped a lot! Thank you!
$endgroup$
– A.Smith
Feb 24 at 15:25
1
1
$begingroup$
That helped a lot! Thank you!
$endgroup$
– A.Smith
Feb 24 at 15:25
$begingroup$
That helped a lot! Thank you!
$endgroup$
– A.Smith
Feb 24 at 15:25
add a comment |
$begingroup$
(Too long for a comment.)
No one else has mentioned this, so I will...
Remember: $y$ is a function of $x$, not something that is constant with respect to variation of $x$. (Forgetting this is a fairly common error when first performing implicit differentiation.) It can help to write "$y(x)$" rather than just "$y$".
Your error is believing $int -x y(x) , mathrmdx = frac-12x^2 y$. But $y$ is not some constant in this integral; it is a function that depends on $x$. Otherwise, what is written could be the absurdity (when, say $y(x) = frac1x^2$), "$int -x cdot frac1x^2 ,mathrmdx = frac-12 x^2 cdot frac1x^2 + C_1 = C$" rather than the correct
$$ int -x cdot frac1x^2 ,mathrmdx = ln|x| + C text. $$
Quick way to verify this: Implicitly differentiate your antiderivative to see if you get the integrand and differential element back:beginalign*
left( frac-12x^2 y(x) right)'
&= frac-12x^2 (y(x))' + left( frac-12x^2 right)' y(x) \
&= frac-12x^2 ,mathrmdy(x) + left( frac-12 cdot 2 x ,mathrmdx right) y(x) \
&= frac-12x^2 ,mathrmdy(x) - x y(x) ,mathrmdx text,
endalign*
which isn't quite "$- x y,mathrmdx$".
(However, we have shown beginalign*
int left( frac-12x^2 fracmathrmdy(x)mathrmdx - x y(x) right) ,mathrmdx
&= int fracmathrmdmathrmdx left( frac-12x^2 y(x) right) , mathrmd x \
&= frac-12x^2 y(x) + C text.
endalign*
Familiarity with this kind of manipulation could be useful in the future.)
$endgroup$
add a comment |
$begingroup$
(Too long for a comment.)
No one else has mentioned this, so I will...
Remember: $y$ is a function of $x$, not something that is constant with respect to variation of $x$. (Forgetting this is a fairly common error when first performing implicit differentiation.) It can help to write "$y(x)$" rather than just "$y$".
Your error is believing $int -x y(x) , mathrmdx = frac-12x^2 y$. But $y$ is not some constant in this integral; it is a function that depends on $x$. Otherwise, what is written could be the absurdity (when, say $y(x) = frac1x^2$), "$int -x cdot frac1x^2 ,mathrmdx = frac-12 x^2 cdot frac1x^2 + C_1 = C$" rather than the correct
$$ int -x cdot frac1x^2 ,mathrmdx = ln|x| + C text. $$
Quick way to verify this: Implicitly differentiate your antiderivative to see if you get the integrand and differential element back:beginalign*
left( frac-12x^2 y(x) right)'
&= frac-12x^2 (y(x))' + left( frac-12x^2 right)' y(x) \
&= frac-12x^2 ,mathrmdy(x) + left( frac-12 cdot 2 x ,mathrmdx right) y(x) \
&= frac-12x^2 ,mathrmdy(x) - x y(x) ,mathrmdx text,
endalign*
which isn't quite "$- x y,mathrmdx$".
(However, we have shown beginalign*
int left( frac-12x^2 fracmathrmdy(x)mathrmdx - x y(x) right) ,mathrmdx
&= int fracmathrmdmathrmdx left( frac-12x^2 y(x) right) , mathrmd x \
&= frac-12x^2 y(x) + C text.
endalign*
Familiarity with this kind of manipulation could be useful in the future.)
$endgroup$
add a comment |
$begingroup$
(Too long for a comment.)
No one else has mentioned this, so I will...
Remember: $y$ is a function of $x$, not something that is constant with respect to variation of $x$. (Forgetting this is a fairly common error when first performing implicit differentiation.) It can help to write "$y(x)$" rather than just "$y$".
Your error is believing $int -x y(x) , mathrmdx = frac-12x^2 y$. But $y$ is not some constant in this integral; it is a function that depends on $x$. Otherwise, what is written could be the absurdity (when, say $y(x) = frac1x^2$), "$int -x cdot frac1x^2 ,mathrmdx = frac-12 x^2 cdot frac1x^2 + C_1 = C$" rather than the correct
$$ int -x cdot frac1x^2 ,mathrmdx = ln|x| + C text. $$
Quick way to verify this: Implicitly differentiate your antiderivative to see if you get the integrand and differential element back:beginalign*
left( frac-12x^2 y(x) right)'
&= frac-12x^2 (y(x))' + left( frac-12x^2 right)' y(x) \
&= frac-12x^2 ,mathrmdy(x) + left( frac-12 cdot 2 x ,mathrmdx right) y(x) \
&= frac-12x^2 ,mathrmdy(x) - x y(x) ,mathrmdx text,
endalign*
which isn't quite "$- x y,mathrmdx$".
(However, we have shown beginalign*
int left( frac-12x^2 fracmathrmdy(x)mathrmdx - x y(x) right) ,mathrmdx
&= int fracmathrmdmathrmdx left( frac-12x^2 y(x) right) , mathrmd x \
&= frac-12x^2 y(x) + C text.
endalign*
Familiarity with this kind of manipulation could be useful in the future.)
$endgroup$
(Too long for a comment.)
No one else has mentioned this, so I will...
Remember: $y$ is a function of $x$, not something that is constant with respect to variation of $x$. (Forgetting this is a fairly common error when first performing implicit differentiation.) It can help to write "$y(x)$" rather than just "$y$".
Your error is believing $int -x y(x) , mathrmdx = frac-12x^2 y$. But $y$ is not some constant in this integral; it is a function that depends on $x$. Otherwise, what is written could be the absurdity (when, say $y(x) = frac1x^2$), "$int -x cdot frac1x^2 ,mathrmdx = frac-12 x^2 cdot frac1x^2 + C_1 = C$" rather than the correct
$$ int -x cdot frac1x^2 ,mathrmdx = ln|x| + C text. $$
Quick way to verify this: Implicitly differentiate your antiderivative to see if you get the integrand and differential element back:beginalign*
left( frac-12x^2 y(x) right)'
&= frac-12x^2 (y(x))' + left( frac-12x^2 right)' y(x) \
&= frac-12x^2 ,mathrmdy(x) + left( frac-12 cdot 2 x ,mathrmdx right) y(x) \
&= frac-12x^2 ,mathrmdy(x) - x y(x) ,mathrmdx text,
endalign*
which isn't quite "$- x y,mathrmdx$".
(However, we have shown beginalign*
int left( frac-12x^2 fracmathrmdy(x)mathrmdx - x y(x) right) ,mathrmdx
&= int fracmathrmdmathrmdx left( frac-12x^2 y(x) right) , mathrmd x \
&= frac-12x^2 y(x) + C text.
endalign*
Familiarity with this kind of manipulation could be useful in the future.)
answered Feb 24 at 19:36
Eric TowersEric Towers
32.9k22370
32.9k22370
add a comment |
add a comment |
$begingroup$
After writing
$$intfracmathrmdyy=int -x,mathrmdx,$$
you get $ln|y|=-fracx^22+c_1$, which implies
$$y(x)=cexp(-x^2/2)quadtextforquad c=pmexp(c_1).$$
After plugging in $y(0)=1$, you get $c=1$, so
$$y(x)=exp(-x^2/2).$$
$endgroup$
$begingroup$
Thank you so so much! I super appreciate it!
$endgroup$
– A.Smith
Feb 24 at 15:53
$begingroup$
You're welcome :)
$endgroup$
– st.math
Feb 24 at 16:32
add a comment |
$begingroup$
After writing
$$intfracmathrmdyy=int -x,mathrmdx,$$
you get $ln|y|=-fracx^22+c_1$, which implies
$$y(x)=cexp(-x^2/2)quadtextforquad c=pmexp(c_1).$$
After plugging in $y(0)=1$, you get $c=1$, so
$$y(x)=exp(-x^2/2).$$
$endgroup$
$begingroup$
Thank you so so much! I super appreciate it!
$endgroup$
– A.Smith
Feb 24 at 15:53
$begingroup$
You're welcome :)
$endgroup$
– st.math
Feb 24 at 16:32
add a comment |
$begingroup$
After writing
$$intfracmathrmdyy=int -x,mathrmdx,$$
you get $ln|y|=-fracx^22+c_1$, which implies
$$y(x)=cexp(-x^2/2)quadtextforquad c=pmexp(c_1).$$
After plugging in $y(0)=1$, you get $c=1$, so
$$y(x)=exp(-x^2/2).$$
$endgroup$
After writing
$$intfracmathrmdyy=int -x,mathrmdx,$$
you get $ln|y|=-fracx^22+c_1$, which implies
$$y(x)=cexp(-x^2/2)quadtextforquad c=pmexp(c_1).$$
After plugging in $y(0)=1$, you get $c=1$, so
$$y(x)=exp(-x^2/2).$$
edited Feb 24 at 10:01
answered Feb 24 at 9:55
st.mathst.math
3818
3818
$begingroup$
Thank you so so much! I super appreciate it!
$endgroup$
– A.Smith
Feb 24 at 15:53
$begingroup$
You're welcome :)
$endgroup$
– st.math
Feb 24 at 16:32
add a comment |
$begingroup$
Thank you so so much! I super appreciate it!
$endgroup$
– A.Smith
Feb 24 at 15:53
$begingroup$
You're welcome :)
$endgroup$
– st.math
Feb 24 at 16:32
$begingroup$
Thank you so so much! I super appreciate it!
$endgroup$
– A.Smith
Feb 24 at 15:53
$begingroup$
Thank you so so much! I super appreciate it!
$endgroup$
– A.Smith
Feb 24 at 15:53
$begingroup$
You're welcome :)
$endgroup$
– st.math
Feb 24 at 16:32
$begingroup$
You're welcome :)
$endgroup$
– st.math
Feb 24 at 16:32
add a comment |
$begingroup$
Here is another method that does not use separation of variables and uses integrating factors. Write
$$
y'+xy=0
$$
and multiply both sides by $e^int x, dx=e^x^2/2$ to get that
$$
0=y'e^x^2/2+xye^x^2/2=fracddx(ye^x^2/2).
$$
So
$$
ye^x^2/2=cimplies y=ce^-x^2/2
$$
for some $c$. Since $y(0)=1$, we deduce that
$$
y=exp(-x^2/2).
$$
$endgroup$
add a comment |
$begingroup$
Here is another method that does not use separation of variables and uses integrating factors. Write
$$
y'+xy=0
$$
and multiply both sides by $e^int x, dx=e^x^2/2$ to get that
$$
0=y'e^x^2/2+xye^x^2/2=fracddx(ye^x^2/2).
$$
So
$$
ye^x^2/2=cimplies y=ce^-x^2/2
$$
for some $c$. Since $y(0)=1$, we deduce that
$$
y=exp(-x^2/2).
$$
$endgroup$
add a comment |
$begingroup$
Here is another method that does not use separation of variables and uses integrating factors. Write
$$
y'+xy=0
$$
and multiply both sides by $e^int x, dx=e^x^2/2$ to get that
$$
0=y'e^x^2/2+xye^x^2/2=fracddx(ye^x^2/2).
$$
So
$$
ye^x^2/2=cimplies y=ce^-x^2/2
$$
for some $c$. Since $y(0)=1$, we deduce that
$$
y=exp(-x^2/2).
$$
$endgroup$
Here is another method that does not use separation of variables and uses integrating factors. Write
$$
y'+xy=0
$$
and multiply both sides by $e^int x, dx=e^x^2/2$ to get that
$$
0=y'e^x^2/2+xye^x^2/2=fracddx(ye^x^2/2).
$$
So
$$
ye^x^2/2=cimplies y=ce^-x^2/2
$$
for some $c$. Since $y(0)=1$, we deduce that
$$
y=exp(-x^2/2).
$$
answered Feb 24 at 16:42
Foobaz JohnFoobaz John
22.8k41452
22.8k41452
add a comment |
add a comment |
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1
$begingroup$
Why don't you separate the variables like $$fracdyy=-xdx?$$ Also the final conclusion is definitely wrong.
$endgroup$
– Fakemistake
Feb 24 at 9:20