1) Here Wolfram Alpha can show me a trend:
Problem:
y' = cos(t) * (2 + cos(t)) / 100
Solution:
y(t) = c_1 + t/200 + sin(t)/50 + 1/200 sin(t) cos(t)

The trend is the linear non-periodic part t/200.

2) But here Wolfram Alpha fails to show me a trend:
Problem (real valued root):
y' = cos(t)^(1/3) * (2 + cos(t)) / 100
Solution:
y(t) = c_1
- 3/200 sqrt(sin^2(t)) cos(t)^(1/3) cot(t)
2F1(1/2, 2/3, 5/3, cos^2(t))
- 3/700 sqrt(sin^2(t)) cos(t) cos(t)^(1/3) cot(t)
2F1(1/2, 7/6, 13/6, cos^2(t))

Doesn't show me a linear non-period part,
but I guess it has one.

Any CAS around that can show me a trend?

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

I can already eliminate FriCAS, at least the default
integrate command doesn't produce a solution at all:

/* Version: FriCAS 1.3.7, WSL2 */

(2) -> integrate(cos(t)^(1/3)*(2+cos(t))/100, t)

t 3+-------+
++ (cos(%A) + 2)\|cos(%A)
(2) | ----------------------- d%A
++ 100

Mild Shock schrieb am Freitag, 20. Oktober 2023 um 22:56:13 UTC+2:

1) Here Wolfram Alpha can show me a trend:
Problem:
y' = cos(t) * (2 + cos(t)) / 100
Solution:
y(t) = c_1 + t/200 + sin(t)/50 + 1/200 sin(t) cos(t)

The trend is the linear non-periodic part t/200.

2) But here Wolfram Alpha fails to show me a trend:
Problem (real valued root):
y' = cos(t)^(1/3) * (2 + cos(t)) / 100
Solution:
y(t) = c_1
- 3/200 sqrt(sin^2(t)) cos(t)^(1/3) cot(t)
2F1(1/2, 2/3, 5/3, cos^2(t))
- 3/700 sqrt(sin^2(t)) cos(t) cos(t)^(1/3) cot(t)
2F1(1/2, 7/6, 13/6, cos^2(t))

Doesn't show me a linear non-period part,
but I guess it has one.

Any CAS around that can show me a trend?

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Saturday, 21 October 2023 at 07:56:13 UTC+11, Mild Shock wrote:

1) Here Wolfram Alpha can show me a trend:
Problem:
y' = cos(t) * (2 + cos(t)) / 100
Solution:
y(t) = c_1 + t/200 + sin(t)/50 + 1/200 sin(t) cos(t)

The trend is the linear non-periodic part t/200.

2) But here Wolfram Alpha fails to show me a trend:
Problem (real valued root):
y' = cos(t)^(1/3) * (2 + cos(t)) / 100
Solution:
y(t) = c_1
- 3/200 sqrt(sin^2(t)) cos(t)^(1/3) cot(t)
2F1(1/2, 2/3, 5/3, cos^2(t))
- 3/700 sqrt(sin^2(t)) cos(t) cos(t)^(1/3) cot(t)
2F1(1/2, 7/6, 13/6, cos^2(t))

Doesn't show me a linear non-period part,
but I guess it has one.

Any CAS around that can show me a trend?

mathHandbook.com can solve eq and auto plot its solution if variable is x, for your example
http://server.drhuang.com/input/?guess=dsolve%28y%27+%3D+cos%28x%29%5E%281%2F3%29+*+%282+%2B+cos%28x%29%29+%29&inp=y%27+%3D+cos%28x%29%5E%281%2F3%29+*+%282+%2B+cos%28x%29%29+&lang=en

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Saturday, 21 October 2023 at 07:56:13 UTC+11, Mild Shock wrote:

1) Here Wolfram Alpha can show me a trend:
Problem:
y' = cos(t) * (2 + cos(t)) / 100
Solution:
y(t) = c_1 + t/200 + sin(t)/50 + 1/200 sin(t) cos(t)

The trend is the linear non-periodic part t/200.

2) But here Wolfram Alpha fails to show me a trend:
Problem (real valued root):
y' = cos(t)^(1/3) * (2 + cos(t)) / 100
Solution:
y(t) = c_1
- 3/200 sqrt(sin^2(t)) cos(t)^(1/3) cot(t)
2F1(1/2, 2/3, 5/3, cos^2(t))
- 3/700 sqrt(sin^2(t)) cos(t) cos(t)^(1/3) cot(t)
2F1(1/2, 7/6, 13/6, cos^2(t))

Doesn't show me a linear non-period part,
but I guess it has one.

Any CAS around that can show me a trend?

Any CAS around that can show me a trend of y'-exp(y)-x-x*x=0 ?

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 11/25/2023 6:03 AM, Dr Huang (DrHuang.com) wrote:

Any CAS around that can show me a trend of y'-exp(y)-x-x*x=0 ?

I have no idea what "trend" of an ode mean. This is a new
term for me that I still have not learned at school and
first time I see it.

But if you just mean the solution of the ode, this is an
easy ode to solve. Just use the substitution u=exp(-y)
which converts it to linear first order ode in u(x) which
is solved using an integrating factor.

Here is my solution attached with slope field plot.

https://12000.org/tmp/solved_problems/nov_25_2023.pdf

The only issue is that there remain unresolved integral.

int( exp(1/3*x^3+1/2*x^2), x)

Which could not be solved by Maple nor Mathematica nor Rubi so the
solution has this unresolved integral in it.



--Nasser

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sunday, 26 November 2023 at 01:13:11 UTC+11, Nasser M. Abbasi wrote:

On 11/25/2023 6:03 AM, Dr Huang (DrHuang.com) wrote:

Any CAS around that can show me a trend of y'-exp(y)-x-x*x=0 ?

I have no idea what "trend" of an ode mean. This is a new
term for me that I still have not learned at school and
first time I see it.

I think it is graphic solution.

But if you just mean the solution of the ode, this is an
easy ode to solve. Just use the substitution u=exp(-y)
which converts it to linear first order ode in u(x) which
is solved using an integrating factor.

Here is my solution attached with slope field plot.

https://12000.org/tmp/solved_problems/nov_25_2023.pdf

The only issue is that there remain unresolved integral.

int( exp(1/3*x^3+1/2*x^2), x)

Which could not be solved by Maple nor Mathematica nor Rubi so the
solution has this unresolved integral in it.

mathHand.com found it with http://server.drhuang.com/input/?guess=dsolve(y(1,x)-exp(y)-x-x^2=0)

--Nasser

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sunday, 26 November 2023 at 14:25:30 UTC+11, Dr Huang (DrHuang.com) wrote:

On Sunday, 26 November 2023 at 01:13:11 UTC+11, Nasser M. Abbasi wrote:

On 11/25/2023 6:03 AM, Dr Huang (DrHuang.com) wrote:

Any CAS around that can show me a trend of y'-exp(y)-x-x*x=0 ?

I have no idea what "trend" of an ode mean. This is a new
term for me that I still have not learned at school and
first time I see it.

I think it is graphic solution.

But if you just mean the solution of the ode, this is an
easy ode to solve. Just use the substitution u=exp(-y)
which converts it to linear first order ode in u(x) which
is solved using an integrating factor.

Here is my solution attached with slope field plot.

https://12000.org/tmp/solved_problems/nov_25_2023.pdf

The only issue is that there remain unresolved integral.

int( exp(1/3*x^3+1/2*x^2), x)

Which could not be solved by Maple nor Mathematica nor Rubi so the
solution has this unresolved integral in it.

mathHand.com found it with http://server.drhuang.com/input/?guess=dsolve(y(1,x)-exp(y)-x-x^2=0) &

http://server.drhuang.com/input/?guess=dsolve(y(1,x)-exp(y)-x-x^2=0)&

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sunday, 26 November 2023 at 14:25:30 UTC+11, Dr Huang (DrHuang.com) wrote:

On Sunday, 26 November 2023 at 01:13:11 UTC+11, Nasser M. Abbasi wrote:

On 11/25/2023 6:03 AM, Dr Huang (DrHuang.com) wrote:

Any CAS around that can show me a trend of y'-exp(y)-x-x*x=0 ?

I have no idea what "trend" of an ode mean. This is a new
term for me that I still have not learned at school and
first time I see it.

I think it is graphic solution.

But if you just mean the solution of the ode, this is an
easy ode to solve. Just use the substitution u=exp(-y)
which converts it to linear first order ode in u(x) which
is solved using an integrating factor.

Here is my solution attached with slope field plot.

https://12000.org/tmp/solved_problems/nov_25_2023.pdf

The only issue is that there remain unresolved integral.

int( exp(1/3*x^3+1/2*x^2), x)

Which could not be solved by Maple nor Mathematica nor Rubi so the
solution has this unresolved integral in it.

mathHand.com found it with http://server.drhuang.com/input/?guess=dsolve(y(1,x)-exp(y)-x-x*x=0) &

http://server.drhuang.com/input/?guess=dsolve(y(1,x)-exp(y)-x-x*x=0)&

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)