restart:
pde := diff(Phi(x,y,z),x)
+
(y^2- a*exp(alpha*x)*(x*y-1))*diff(Phi(x,y,z),y)
+
(c*exp(beta*x)*z^2+b*exp(-beta*x))*diff(Phi(x,y,z),z)
=
0;
LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJFBoaUc2IjYlSSJ4R0YqSSJ5R0YqSSJ6R0YqRiwiIiIqJiwmKiRGLSIiI0YvKihJImFHRipGLy1JJGV4cEc2JEYmSShfc3lzbGliR0YqNiMqJkkmYWxwaGFHRipGL0YsRi9GLywmKiZGLEYvRi1GL0YvISIiRi9GL0Y/Ri8tRiU2JEYoRi1GL0YvKiYsJiooSSJjR0YqRi8tRjc2IyomSSViZXRhR0YqRi9GLEYvRi9GLkYzRi8qJkkiYkdGKkYvLUY3NiMsJEZIRj9GL0YvRi8tRiU2JEYoRi5GL0YvIiIh
# Write the pde as F . grad(Phi) = 0 where F is the vector (A, B, C)
# Next parameterize F as a function of s : F(s) = (A(s), B(s), C(s))
A := s -> 1;
B := s -> (y(s)^2- a*exp(alpha*x(s))*(x(s)*y(s)-1));
C := s -> (c*exp(beta*x(s))*z(s)^2+b*exp(-beta*x(s)));
Zio2I0kic0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlIiIiRiVGJUYl
Zio2I0kic0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJC1JInlHRiU2IzkkIiIjIiIiKihJImFHRiVGMC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJJmFscGhhR0YlRjAtSSJ4R0YlRi1GMEYwLCYqJkY7RjBGK0YwRjAhIiJGMEYwRj9GJUYlRiU=
Zio2I0kic0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqKEkiY0dGJSIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJJWJldGFHRiVGLC1JInhHRiU2IzkkRixGLC1JInpHRiVGNyIiI0YsKiZJImJHRiVGLC1GLjYjLCRGMyEiIkYsRixGJUYlRiU=
# Define the characteristic curve associated to the pde as the curve defined by (U(s), V(s), W(s))
# where :
# diff(U(s), s) = A(s)
# diff(V(s), s) = B(s)
# diff(W(s), s) = C(s)
#
# Then Phi(s) is constant along each characteristic curve
#
# Characteristic curve, component 1
eq1 := diff(U(s), s) = A(s);
dsolve(eq1, U(s)):
U := unapply(rhs(%), s);
Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJVRzYiNiNJInNHRilGKyIiIg==
Zio2I0kic0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIkkkX0MxR0YlRitGJUYlRiU=
# Characteristic curve, component 3
eq3 := subs({x(s)=U(s), z(s)=W(s)}, diff(W(s), s) = C(s));
dsolve(eq3, W(s))
Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJXRzYiNiNJInNHRilGKywmKihJImNHRikiIiItSSRleHBHNiRGJUkoX3N5c2xpYkdGKTYjKiZJJWJldGFHRilGLywmRitGL0kkX0MxR0YpRi9GL0YvRiciIiNGLyomSSJiR0YpRi8tRjE2IywkRjUhIiJGL0Yv
Ly1JIldHNiI2I0kic0dGJSwkKiosJiooLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJkklYmV0YUdGJSIiIkkkX0MxR0YlRjRGNC1GLTYjKiZGM0Y0RidGNEY0RjNGNEY0KiYtSSR0YW5HRi42IywkKioqKEYsIiIjRjZGQCwmKiZJImJHRiVGNEkiY0dGJUY0IiIlKiRGM0ZAISIiRjQjRjRGQCwmSSRfQzJHRiVGNEYnRkdGNEYsRkdGNkZHRkhGNEY/RkhGNEY0RiwhIiNGNkZLRkRGRyNGR0ZA
# Characteristic curve, component 2
eq2 := subs({x(s)=U(s), y(s)=V(s)}, diff(V(s), s) = expand(B(s)));
infolevel[dsolve] := 4:
dsolve(eq2, V(s))
Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJWRzYiNiNJInNHRilGKywoKiRGJyIiIyIiIioqSSJhR0YpRi8tSSRleHBHNiRGJUkoX3N5c2xpYkdGKTYjKiZJJmFscGhhR0YpRi8sJkYrRi9JJF9DMUdGKUYvRi9GL0Y5Ri9GJ0YvISIiKiZGMUYvRjJGL0Yv
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
trying Riccati
trying inverse_Riccati
trying 1st order ODE linearizable_by_differentiation
--- Trying Lie symmetry methods, 1st order ---
-> Computing symmetries using: way = 4
-> Computing symmetries using: way = 2
-> Computing symmetries using: way = 6
trying symmetry patterns for 1st order ODEs
-> trying a symmetry pattern of the form [F(x)*G(y), 0]
-> trying a symmetry pattern of the form [0, F(x)*G(y)]
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)]
-> Computing symmetries using: way = HINT
-> Calling odsolve with the ODE diff(y(x) x) = 2*y(x)/x y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x) = y(x)/x y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)+y(x)*alpha y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)+exp(alpha*(x+_C1))*a*K[1]*(x+_C1) y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
<- quadrature successful
-> Calling odsolve with the ODE diff(y(x) x)+(y(x)*_C1*alpha+y(x)*alpha*x+y(x)+2*K[1])/(x+_C1) y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)+K[1] y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)+y(x)*alpha-K[1] y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)+y(x)*(_C1*alpha+alpha*x+1)/(x+_C1) y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x) = 0 y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x) = -y(x)*alpha y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)-2*y(x)/x y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)-(x*alpha*K[1]+y(x))/x y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)-(x*_C1*alpha*K[1]+y(x)*_C1+x*K[1]-K[1]*alpha)/(_C1*x-1) y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)-y(x)/x y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)-y(x)*_C1/(_C1*x-1) y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> Calling odsolve with the ODE diff(y(x) x)+(exp(alpha*_C1)*a*K[1]-2*y(x))/x y(x)
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
-> trying a symmetry pattern of the form [F(x),G(x)]
-> trying a symmetry pattern of the form [F(y),G(y)]
-> trying a symmetry pattern of the form [F(x)+G(y), 0]
-> trying a symmetry pattern of the form [0, F(x)+G(y)]
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)]
-> trying a symmetry pattern of conformal type
# Note there is no explicit solution found: I suppose this is the reason why Maple cannot solve the pde
eq2 := subs({x(s)=U(s), y(s)=V(s), alpha=0}, diff(V(s), s) = expand(B(s)));
infolevel[dsolve] := 4:
dsolve(eq2, V(s))
Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJWRzYiNiNJInNHRilGKywoKiRGJyIiIyIiIioqSSJhR0YpRi8tSSRleHBHNiRGJUkoX3N5c2xpYkdGKTYjIiIhRi8sJkYrRi9JJF9DMUdGKUYvRi9GJ0YvISIiKiZGMUYvRjJGL0Yv
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
Chini's absolute invariant is: a*(s+_C1)^2
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
Found potential symmetries: [0 1] [exp(-V)/a exp(-V)*(s+_C1)]
Proceeding with integration step.
Ly1JIlZHNiI2I0kic0dGJSwkKiosKCooSSRfQzFHRiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooRidGLUYsRi1JImFHRiVGLSEiIi1GLzYjLCQqJkY1Ri1GJyIiIyNGNkY7Ri1GNiooRidGLUYuRjZGN0YtRjYqJCwmKigtRi82IywkKiZGLEY7RjVGLUY8Ri0tSSRlcmZHRjA2IywkKiYsJEY1ISIjI0YtRjssJkYnRi1GLEYtRi1GTEYtLCQqJkkjUGlHRjFGLUY1Ri1GS0ZMRjxJJF9DMkdGJUYtRjZGLUYtRi5GLUY3RjZGNUYtRjY=
pdsolve(subs(alpha=0, pde), Phi(x,y,z))
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
Chini's absolute invariant is: a*x^2
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
Found potential symmetries: [0 1] [exp(-y) exp(-y)*a*x]
Proceeding with integration step.
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
Chini's absolute invariant is: beta^2/(c*b)
<- Chini successful
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
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