6.8.5 3.1

6.8.5.1 [1786] Problem 1

problem number 1786

Added July 1, 2019.

Problem Chapter 8.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a e^{\lambda x} w_y + b e^{\beta x} w_z = c e^{\gamma x} w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*x]*D[w[x,y,z],z]== c*Exp[gamma*x]*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\gamma x}}{\gamma }} c_1\left (y-\frac {a e^{\lambda x}}{\lambda },z-\frac {b e^{\beta x}}{\beta }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*exp(beta*x)*diff(w(x,y,z),z)= c*exp(gamma*x)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+y \lambda }{\lambda }, \frac {-b \,{\mathrm e}^{\beta x}+z \beta }{\beta }\right ) {\mathrm e}^{\frac {c \,{\mathrm e}^{\gamma x}}{\gamma }}\]

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6.8.5.2 [1787] Problem 2

problem number 1787

Added July 1, 2019.

Problem Chapter 8.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a e^{\lambda x} w_y + b e^{\beta x} w_z = (c e^{\gamma y}+s e^{\mu z}) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*x]*D[w[x,y,z],z]== (c*Exp[gamma*y]+s Exp[mu*z])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },z-\frac {b e^{\beta x}}{\beta }\right ) \exp \left (\frac {c \operatorname {ExpIntegralEi}\left (\frac {a e^{\lambda x} \gamma }{\lambda }\right ) e^{\gamma \left (y-\frac {a e^{\lambda x}}{\lambda }\right )}}{\lambda }+\frac {s \operatorname {ExpIntegralEi}\left (\frac {b e^{\beta x} \mu }{\beta }\right ) e^{\mu \left (z-\frac {b e^{\beta x}}{\beta }\right )}}{\beta }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*exp(beta*x)*diff(w(x,y,z),z)= (c*exp(gamma*y)+s*exp(mu*z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+y \lambda }{\lambda }, \frac {z \beta -b \,{\mathrm e}^{\beta x}}{\beta }\right ) {\mathrm e}^{-\frac {c \,{\mathrm e}^{-\frac {\gamma \left (a \,{\mathrm e}^{\lambda x}-y \lambda \right )}{\lambda }} \operatorname {Ei}_{1}\left (-\frac {\gamma a \,{\mathrm e}^{\lambda x}}{\lambda }\right )}{\lambda }-\frac {s \,{\mathrm e}^{\frac {\mu \left (z \beta -b \,{\mathrm e}^{\beta x}\right )}{\beta }} \operatorname {Ei}_{1}\left (-\frac {\mu b \,{\mathrm e}^{\beta x}}{\beta }\right )}{\beta }}\]

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6.8.5.3 [1788] Problem 3

problem number 1788

Added July 2, 2019.

Problem Chapter 8.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a e^{\lambda y} w_y + b e^{\beta y} w_z = (c e^{\gamma x}+s e^{\mu z}) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*y]*D[w[x, y,z], y] +b*Exp[beta*y]*D[w[x,y,z],z]== (c*Exp[gamma*x]+s Exp[mu*z])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a \lambda x+e^{-\lambda y}}{\lambda },\frac {b \left (e^{-\lambda y}\right )^{1-\frac {\beta }{\lambda }}}{a (\lambda -\beta )}+z\right ) \exp \left (\int _1^x\left (e^{\gamma K[1]} c+\exp \left (-\frac {\mu \left (b \lambda (x-K[1]) \left (a \lambda (x-K[1])+e^{-\lambda y}\right )^{-\frac {\beta }{\lambda }}+(\beta -\lambda ) z+\frac {b e^{-\lambda y} \left (\left (a \lambda (x-K[1])+e^{-\lambda y}\right )^{-\frac {\beta }{\lambda }}-\left (e^{-\lambda y}\right )^{-\frac {\beta }{\lambda }}\right )}{a}\right )}{\lambda -\beta }\right ) s\right )dK[1]\right )\right \}\right \}\]

Generates Solve::incnst: Inconsistent or redundant transcendental equation

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+a*exp(lambda*y)*diff(w(x,y,z),y)+b*exp(beta*y)*diff(w(x,y,z),z)= (c*exp(gamma*x)+s*exp(mu*z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \lambda x -{\mathrm e}^{-\lambda y}}{a \lambda }, \frac {-\left ({\mathrm e}^{\lambda y}\right )^{\frac {\beta }{\lambda }} {\mathrm e}^{-\lambda y} b +a z \left (\beta -\lambda \right )}{a \left (\beta -\lambda \right )}\right ) {\mathrm e}^{\int _{}^{x}\left (c \,{\mathrm e}^{\gamma \textit {\_a}}+s \,{\mathrm e}^{\frac {\left (\left ({\mathrm e}^{-\lambda y}+a \lambda \left (x -\textit {\_a} \right )\right ) b \left (\frac {1}{{\mathrm e}^{-\lambda y}+a \lambda \left (x -\textit {\_a} \right )}\right )^{\frac {\beta }{\lambda }}-\left ({\mathrm e}^{\lambda y}\right )^{\frac {\beta }{\lambda }} {\mathrm e}^{-\lambda y} b +a z \left (\beta -\lambda \right )\right ) \mu }{a \left (\beta -\lambda \right )}}\right )d \textit {\_a}}\]

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6.8.5.4 [1789] Problem 4

problem number 1789

Added July 2, 2019.

Problem Chapter 8.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + (A_1 e^{\alpha _1 x} + B_1 e^{\nu _1 x+\lambda y}) w_y + (A_2 e^{\alpha _2 x} + B_2 e^{\nu _2 x+\beta z}) w_z = k e^{\gamma z} w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (A1*Exp[alpha1*x] + B1*Exp[nu1*x+lambda*y])*D[w[x, y,z], y] + (A2*Exp[alpha2*x] + B2*Exp[nu2*x+beta*z])*D[w[x,y,z],z]== k*Exp[gamma*x]*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+(A1*exp(alpha1*x) + B1*exp(nu1*x+lambda*y))*diff(w(x,y,z),y)+(A2*exp(alpha2*x) + B2*exp(nu2*x+beta*z))*diff(w(x,y,z),z)= k*exp(gamma*x)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-{\mathrm e}^{\frac {\lambda \left (\operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} x}-\alpha \operatorname {1} y \right )}{\alpha \operatorname {1}}}-\operatorname {B1} \int {\mathrm e}^{\frac {\lambda \operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} x}+\nu \operatorname {1} x \alpha \operatorname {1} }{\alpha \operatorname {1}}}d x \lambda }{\lambda }, \frac {-{\mathrm e}^{\frac {\beta \left (\operatorname {A2} \,{\mathrm e}^{\alpha \operatorname {2} x}-\alpha \operatorname {2} z \right )}{\alpha \operatorname {2}}}-\operatorname {B2} \int {\mathrm e}^{\frac {\beta \operatorname {A2} \,{\mathrm e}^{\alpha \operatorname {2} x}+\nu \operatorname {2} x \alpha \operatorname {2} }{\alpha \operatorname {2}}}d x \beta }{\beta }\right ) {\mathrm e}^{\frac {k \,{\mathrm e}^{\gamma x}}{\gamma }}\]

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6.8.5.5 [1790] Problem 5

problem number 1790

Added July 2, 2019.

Problem Chapter 8.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a e^{\alpha x} w_x + b e^{\beta y} w_y + c e^{\gamma z} w_z = k e^{\lambda x} w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[alpha*x]*D[w[x, y,z], x] + b*Exp[beta*y]*D[w[x, y,z], y] + c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {k e^{x (\lambda -\alpha )}}{a (\lambda -\alpha )}} c_1\left (\frac {b e^{-\alpha x}}{a \alpha }-\frac {e^{-\beta y}}{\beta },\frac {c e^{-\alpha x}}{a \alpha }-\frac {e^{-\gamma z}}{\gamma }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  a*exp(alpha*x)*diff(w(x,y,z),x)+b*exp(beta*y)*diff(w(x,y,z),y)+c*exp(gamma*z)*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-{\mathrm e}^{-\beta y} a \alpha +{\mathrm e}^{-\alpha x} \beta b}{\alpha b \beta }, \frac {-{\mathrm e}^{-\gamma z} a \alpha +{\mathrm e}^{-\alpha x} \gamma c}{\alpha c \gamma }\right ) {\mathrm e}^{-\frac {k \,{\mathrm e}^{x \left (-\alpha +\lambda \right )}}{\left (\alpha -\lambda \right ) a}}\]

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6.8.5.6 [1791] Problem 6

problem number 1791

Added July 2, 2019.

Problem Chapter 8.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a e^{\beta y} w_x + b e^{\alpha x} w_y + c e^{\gamma z} w_z = k e^{\lambda x} w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Exp[beta*y]*D[w[x, y,z], x] + b*Exp[alpha*x]*D[w[x, y,z], y] + c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a*exp(beta*y)*diff(w(x,y,z),x)+b*exp(alpha*x)*diff(w(x,y,z),y)+c*exp(gamma*z)*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {{\mathrm e}^{\beta y} a \alpha -{\mathrm e}^{\alpha x} \beta b}{\alpha \beta b}, -\frac {b \left (-\ln \left (\frac {{\mathrm e}^{\beta y} a \alpha }{\beta b}\right ) c \gamma +{\mathrm e}^{-\gamma z} \left ({\mathrm e}^{\beta y} a \alpha -{\mathrm e}^{\alpha x} \beta b \right )+\alpha c \gamma x \right ) \beta }{\left ({\mathrm e}^{\beta y} a \alpha -{\mathrm e}^{\alpha x} \beta b \right ) \alpha c \gamma }\right ) {\mathrm e}^{k \alpha \int _{}^{x}\frac {{\mathrm e}^{\lambda \textit {\_a}}}{{\mathrm e}^{\alpha \textit {\_a}} \beta b +{\mathrm e}^{\beta y} a \alpha -{\mathrm e}^{\alpha x} \beta b}d \textit {\_a}}\]

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6.8.5.7 [1792] Problem 7

problem number 1792

Added July 2, 2019.

Problem Chapter 8.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ (a_1+a_2 e^{\alpha x}) w_x + (b_1+b_2 e^{\beta y}) w_y + (c_1+c_2 e^{\gamma z}) w_z = (k_1+k_2 e^{\alpha x}) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + (b1+b2*Exp[beta*y])*D[w[x, y,z], y] + (c1+c2*Exp[gamma*z])*D[w[x,y,z],z]== (k1+k2*Exp[alpha*x])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \left (e^{\alpha x}\right )^{\frac {\text {k1}}{\text {a1} \alpha }} \left (\text {a1} \alpha \left (\text {a1}+\text {a2} e^{\alpha x}\right )\right )^{\frac {\text {a1} \text {k2}-\text {a2} \text {k1}}{\text {a1} \text {a2} \alpha }} c_1\left (-\frac {\log \left (\text {b1} \beta \left (\text {b1}+\text {b2} e^{\beta y}\right ) e^{\frac {\text {b1} \beta x}{\text {a1}}-\beta y} \left (\text {a1} \alpha \left (\text {a1}+\text {a2} e^{\alpha x}\right )\right )^{-\frac {\text {b1} \beta }{\text {a1} \alpha }}\right )}{\text {b1} \beta },-\frac {\log \left (\text {c1} \gamma \left (\text {c1}+\text {c2} e^{\gamma z}\right ) e^{\frac {\text {c1} \gamma x}{\text {a1}}-\gamma z} \left (\text {a1} \alpha \left (\text {a1}+\text {a2} e^{\alpha x}\right )\right )^{-\frac {\text {c1} \gamma }{\text {a1} \alpha }}\right )}{\text {c1} \gamma }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  (a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+(b1+b2*exp(beta*y))*diff(w(x,y,z),y)+(c1+c2*exp(gamma*z))*diff(w(x,y,z),z)= (k1+k2*exp(alpha*x))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\operatorname {a1} \operatorname {k2} -\operatorname {a2} \operatorname {k1}}{\alpha \operatorname {a1} \operatorname {a2}}} f_{1} \left (\frac {-x \beta \operatorname {b1} +\ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\beta \operatorname {b1}}{\alpha \operatorname {a1}}} \left (-\operatorname {b1} +\operatorname {RootOf}\left (y \alpha \operatorname {a1} \beta -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\beta \operatorname {b1}}{\alpha \operatorname {a1}}} \left (-\operatorname {b1} +\textit {\_Z} \right )}{\operatorname {b2}}\right )+\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \beta \operatorname {b1} \right )\right )}{\operatorname {b2} \operatorname {RootOf}\left (y \alpha \operatorname {a1} \beta -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\beta \operatorname {b1}}{\alpha \operatorname {a1}}} \left (-\operatorname {b1} +\textit {\_Z} \right )}{\operatorname {b2}}\right )+\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \beta \operatorname {b1} \right )}\right ) \operatorname {a1}}{\operatorname {a1} \operatorname {b1} \beta }, \frac {-\gamma \operatorname {c1} x +\ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\gamma \operatorname {c1}}{\alpha \operatorname {a1}}} \left (-\operatorname {c1} +\operatorname {RootOf}\left (z \alpha \operatorname {a1} \gamma -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\gamma \operatorname {c1}}{\alpha \operatorname {a1}}} \left (-\operatorname {c1} +\textit {\_Z} \right )}{\operatorname {c2}}\right )+\gamma \operatorname {c1} \ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )\right )\right )}{\operatorname {c2} \operatorname {RootOf}\left (z \alpha \operatorname {a1} \gamma -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\gamma \operatorname {c1}}{\alpha \operatorname {a1}}} \left (-\operatorname {c1} +\textit {\_Z} \right )}{\operatorname {c2}}\right )+\gamma \operatorname {c1} \ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )\right )}\right ) \operatorname {a1}}{\operatorname {a1} \operatorname {c1} \gamma }\right ) \left ({\mathrm e}^{\alpha x}\right )^{\frac {\operatorname {k1}}{\alpha \operatorname {a1}}}\]

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6.8.5.8 [1793] Problem 8

problem number 1793

Added July 2, 2019.

Problem Chapter 8.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ e^{\beta y} (a_1+a_2 e^{\alpha x}) w_x + e^{\alpha x} (b_1+b_2 e^{\beta y}) w_y + c e^{\beta y+\gamma z} w_z = k_3 e^{\beta y} (k_1+k_2 e^{\alpha x}) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  Exp[beta*y]*(a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + Exp[alpha*x]*(b1+b2*Exp[beta*y])*D[w[x, y,z], y] + c*Exp[beta*y+gamma*z]*D[w[x,y,z],z]== k3*Exp[beta*y]*(k1+k2*Exp[alpha*x])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {\text {k3} \left ((\text {a1} \text {k2}-\text {a2} \text {k1}) \log \left (\text {a1} \alpha \left (\text {a1}+\text {a2} e^{\alpha x}\right )\right )+\text {a2} \text {k1} \log \left (e^{\alpha x}\right )\right )}{\text {a1} \text {a2} \alpha }\right ) c_1\left (\frac {c \log \left (\text {a1} \alpha \left (\text {a1}+\text {a2} e^{\alpha x}\right )\right )}{\text {a1} \alpha }-\frac {c x}{\text {a1}}-\frac {e^{-\gamma z}}{\gamma },\frac {\log \left (\text {b1}+\text {b2} e^{\beta y}\right )}{\text {b2} \beta }-\frac {\log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a2} \alpha }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  exp(beta*y)*(a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+exp(alpha*x)*(b1+b2*exp(beta*y))*diff(w(x,y,z),y)+c*exp(beta*y+gamma*z)*diff(w(x,y,z),z)= k3*exp(beta*y)*(k1+k2*exp(alpha*x))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\operatorname {k3} \left (\operatorname {a1} \operatorname {k2} -\operatorname {a2} \operatorname {k1} \right )}{\alpha \operatorname {a1} \operatorname {a2}}} \left ({\mathrm e}^{\alpha x}\right )^{\frac {\operatorname {k3} \operatorname {k1}}{\alpha \operatorname {a1}}} f_{1} \left (\frac {-x \beta \operatorname {b2} +\ln \left (\frac {\operatorname {RootOf}\left (y \alpha \operatorname {a2} \beta +\alpha x \beta \operatorname {b2} -\alpha \operatorname {a2} \ln \left (\frac {\operatorname {b1} \left (\operatorname {a2} +{\mathrm e}^{-\alpha x} \operatorname {a1} \right )^{-\frac {\beta \operatorname {b2}}{\alpha \operatorname {a2}}}}{-\operatorname {b2} +\textit {\_Z}}\right )-\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \operatorname {b2} \beta \right ) \operatorname {b1} \left (\operatorname {a2} +{\mathrm e}^{-\alpha x} \operatorname {a1} \right )^{-\frac {\beta \operatorname {b2}}{\alpha \operatorname {a2}}}}{-\operatorname {b2} +\operatorname {RootOf}\left (y \alpha \operatorname {a2} \beta +\alpha x \beta \operatorname {b2} -\alpha \operatorname {a2} \ln \left (\frac {\operatorname {b1} \left (\operatorname {a2} +{\mathrm e}^{-\alpha x} \operatorname {a1} \right )^{-\frac {\beta \operatorname {b2}}{\alpha \operatorname {a2}}}}{-\operatorname {b2} +\textit {\_Z}}\right )-\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \operatorname {b2} \beta \right )}\right ) \operatorname {a2}}{\operatorname {a2} \operatorname {b2} \beta }, \frac {c \gamma \left (\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )-\alpha x \right )-{\mathrm e}^{-\gamma z} \alpha \operatorname {a1}}{\operatorname {a1} \alpha c \gamma }\right )\]

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