what is solution of y''-y' y-2x=0 and how to test it by online software? it is seems wolfram solution is wrong?
mathHand.com
On 1/27/2021 5:05 PM, drhu...@gmail.com wrote:
what is solution of y''-y' y-2x=0 and how to test it by online software?
it is seems wolfram solution is wrong?
mathHand.com
I do not try Wolfram alpha, but Mathematica solves this. I assume
answer it gives is correct.
Here is how to solve this by hand.
Integrate both sides w.r.t. x
int ( y''-y' y ,x) = int( 2 x,x)
-y^2/2 + y' = x^2 + C
This is first order ode
y' = C + x^2 + y^2/2
Compare to https://en.wikipedia.org/wiki/Riccati_equation shows it is RICCATI
y' = q0(x) + q2(x) y^2
with missing q1(x). Here q0=c+x^2 abd q2(x) = 1/2.
This has standard method of solving as shown in the above wiki page.
--Nasser
On Thursday, 28 January 2021 at 13:01:26 UTC+11, Nasser M. Abbasi wrote:
On 1/27/2021 5:05 PM, drhu...@gmail.com wrote:
what is solution of y''-y' y-2x=0 and how to test it by online software? >>> it is seems wolfram solution is wrong?I do not try Wolfram alpha, but Mathematica solves this. I assume
mathHand.com
answer it gives is correct.
Here is how to solve this by hand.
Integrate both sides w.r.t. x
int ( y''-y' y ,x) = int( 2 x,x)
-y^2/2 + y' = x^2 + C
This is first order ode
y' = C + x^2 + y^2/2
Compare to https://en.wikipedia.org/wiki/Riccati_equation shows it is RICCATI
y' = q0(x) + q2(x) y^2
with missing q1(x). Here q0=c+x^2 abd q2(x) = 1/2.
This has standard method of solving as shown in the above wiki page.
--Nasser
if you put the solution of the Riccati equation back to the y''-y' y-2x=0, you got nonzero, so it is wrong.
mathHand.com gives http://server.drhuang.com/input/?guess=dsolve%28++ds%28y%2Cx%2C2%29-++ds%28y%29*y-2x%29&inp=++ds%28y%2Cx%2C2%29-++ds%28y%29*y-2x
On Thursday, 28 January 2021 at 13:01:26 UTC+11, Nasser M. Abbasi wrote:
On 1/27/2021 5:05 PM, drhu...@gmail.com wrote:
what is solution of y''-y' y-2x=0 and how to test it by online software? >>> it is seems wolfram solution is wrong?I do not try Wolfram alpha, but Mathematica solves this. I assume
mathHand.com
answer it gives is correct.
Here is how to solve this by hand.
Integrate both sides w.r.t. x
int ( y''-y' y ,x) = int( 2 x,x)
-y^2/2 + y' = x^2 + C
This is first order ode
y' = C + x^2 + y^2/2
Compare to https://en.wikipedia.org/wiki/Riccati_equation shows it is RICCATI
y' = q0(x) + q2(x) y^2
with missing q1(x). Here q0=c+x^2 abd q2(x) = 1/2.
This has standard method of solving as shown in the above wiki page.
--Nasser
if you put the solution of the Riccati equation back to the y''-y' y-2x=0, you got nonzero, so it is wrong.
mathHand.com gives http://server.drhuang.com/input/?guess=dsolve%28++ds%28y%2Cx%2C2%29-++ds%28y%29*y-2x%29&inp=++ds%28y%2Cx%2C2%29-++ds%28y%29*y-2x
The solution in y(x) is obtained from this u(x) using the
transformation y= -u'/(2 u).
On 1/31/2021 6:29 AM, drhu...@gmail.com wrote:
On Thursday, 28 January 2021 at 13:01:26 UTC+11, Nasser M. Abbasi wrote:
On 1/27/2021 5:05 PM, drhu...@gmail.com wrote:
what is solution of y''-y' y-2x=0 and how to test it by online software? >>> it is seems wolfram solution is wrong?I do not try Wolfram alpha, but Mathematica solves this. I assume
mathHand.com
answer it gives is correct.
Here is how to solve this by hand.
Integrate both sides w.r.t. x
int ( y''-y' y ,x) = int( 2 x,x)
-y^2/2 + y' = x^2 + C
This is first order ode
y' = C + x^2 + y^2/2
Compare to https://en.wikipedia.org/wiki/Riccati_equation shows it is RICCATI
y' = q0(x) + q2(x) y^2
with missing q1(x). Here q0=c+x^2 abd q2(x) = 1/2.
This has standard method of solving as shown in the above wiki page.
--Nasser
if you put the solution of the Riccati equation back to the y''-y' y-2x=0, you got nonzero, so it is wrong.
mathHand.com gives http://server.drhuang.com/input/?guess=dsolve%28++ds%28y%2Cx%2C2%29-++ds%28y%29*y-2x%29&inp=++ds%28y%2Cx%2C2%29-++ds%28y%29*y-2x
Well, it works for me. I solved it and put the solution here
https://www.12000.org/my_notes/solving_ODE/current_version/insu3827.htm
I used Maple to solve the generated riccati ODE for now.
Maple verified the solution.
=====================
ode:=diff(y(x),x$2)-diff(y(x),x)*y(x)-2*x=0;
mysol:=y(x) = (_C3*(sqrt(2)*I - 6)*WhittakerM(-sqrt(2)*I/8 + 1, 1/4, I/2*sqrt(2)*x^2)
+ 8*WhittakerW(-sqrt(2)*I/8 + 1, 1/4, I/2*sqrt(2)*x^2)*_C4
- 2*(_C4*WhittakerW(-I/8*sqrt(2), 1/4, I/2*sqrt(2)*x^2)
+ _C3*WhittakerM(-I/8*sqrt(2), 1/4, I/2*sqrt(2)*x^2))*
(-1 + (x^2*I + I/2)*sqrt(2)))/(2*x*(_C4*WhittakerW(-I/8*sqrt(2), 1/4, I/2*sqrt(2)*x^2)
+ _C3*WhittakerM(-I/8*sqrt(2), 1/4, I/2*sqrt(2)*x^2)));
odetest(mysol,ode)
0
===========================
Zero means correct solution.
Since you did not show what you, and just said it is wrong, it is
hard for someone to say what the issue is with what you did.
Sysop: | Keyop |
---|---|
Location: | Huddersfield, West Yorkshire, UK |
Users: | 112 |
Nodes: | 8 (1 / 7) |
Uptime: | 244:32:24 |
Calls: | 2,467 |
Calls today: | 1 |
Files: | 8,614 |
Messages: | 1,885,831 |