Dear Jay
The DE is given in exact form
2xsin(3y)+3x^2cos(3y)dy=0
which is in exact DE form [M(x,y)dx+N(x,y)dy] and can easily shown to be
2xsin(3y)dx+3x^2cos(3y)dy=d[x^2*sin(3y)]=0 with simple solution
x^2*sin(3y) = constant
OK trying this in Maxima [setting up for desolve] we rearrange the DE as
dy/dx = -2x*sin(3y)/(3x^2*cos(3y)) = -2tan(3y)/(3x)
and enter the DE into Maxima as follows:
(C3) de1:'diff(y(x),x)=-2*tan(3*y)/(3*x);
d 2 TAN(3 y)
(D3) -- (y(x)) = - ----------
dx 3 x
(C4) desolve(de1,y(x));
1
2 LAPLACE(-, x, LVAR) TAN(3 y) - 3 y(0)
x
(D4) y(x) = ILT(- ---------------------------------------, LVAR, x)
or using the classical calling convention [as stated in the manual]:
3 LVAR
(C5) desolve([D3],[y(x)]);
1
2 LAPLACE(-, x, LVAR) TAN(3 y) - 3 y(0)
x
(D5) y(x) = ILT(- ---------------------------------------, LVAR, x)
3 LVAR
Looks like I may have made a mistake setting up the DE not having y(x) instead of y on RHS: trying again:
(C6) de1:'diff(y(x),x)=-2*tan(3*y(x))/(3*x);
d 2 TAN(3 y(x))
(D6) -- (y(x)) = - -------------
dx 3 x
(C7) desolve(de1,y(x));
(D7) y(x) =
INF
/
[
2 I LAPLACE(TAN(3 y(x)), x, LVAR1) dLVAR1 - 3 y(0)
]
/
LVAR
ILT(- ------------------------------------------------------, LVAR, x)
3 LVAR
OR
(C10) desolve(D6,y(x));
(D10) y(x) =
INF
/
[
2 I LAPLACE(TAN(3 y(x)), x, LVAR1) dLVAR1 - 3 y(0)
]
/
LVAR
ILT(- ------------------------------------------------------, LVAR, x)
3 LVAR
NOTE: I checked I was doing this right by trying an even simpler DE
dy/dx = y with simple solution y = A*e^x
(C8) de2:y(x)='diff(y(x),x);
d
(D8) y(x) = -- (y(x))
dx
(C9) desolve(de2,y(x));
x
(D9) y(x) = y(0) %E
Looks OK.
Note: I tried the DE in various forms and found that DESOLVE couldn't even get an answer:
(C11) de3:2*x*sin(3*y(x))+3*x^2*cos(3*y(x))*'diff(y(x),x)=0;
2 d
(D11) 2 x SIN(3 y(x)) + 3 x (-- (y(x))) COS(3 y(x)) = 0
dx
(C12) desolve(de3,y(x));
DESOLVE can't handle this case.
-- an error. Quitting. To debug this try DEBUGMODE(TRUE);)
(C13) de4:'diff(y(x),x)*cos(3*y(x))*3*x^2+2*x*sin(3*y(x))=0;
2 d
(D13) 2 x SIN(3 y(x)) + 3 x (-- (y(x))) COS(3 y(x)) = 0
dx
(C14) desolve(de4,y(x));
DESOLVE can't handle this case.
-- an error. Quitting. To debug this try DEBUGMODE(TRUE);)
(C15) de5:'diff(y(x),x)*cos(3*y(x))=-2*sin(3*y(x))/3/x;
d 2 SIN(3 y(x))
(D15) (-- (y(x))) COS(3 y(x)) = - -------------
dx 3 x
(C16) desolve(de5,y(x));
DESOLVE can't handle this case.
-- an error. Quitting. To debug this try DEBUGMODE(TRUE);)
Finally making dy/dx the subject on the LHS did yield the same unhelpful solution [with sin/cos instead of tan] as previous:
(C17) de6:'diff(y(x),x)=-2*sin(3*y(x))/3/x/cos(3*y(x));
d 2 SIN(3 y(x))
(D17) -- (y(x)) = - ---------------
dx 3 x COS(3 y(x))
(C18) desolve(de6,y(x));
(D18) y(x) =
INF
/
[ SIN(3 y(x))
2 I LAPLACE(-----------, x, LVAR1) dLVAR1 - 3 y(0)
] COS(3 y(x))
/
LVAR
ILT(- ------------------------------------------------------, LVAR, x)
3 LVAR
kind regards
Stephen
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