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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+\lambda y}{\lambda }, \frac {-b \,{\mathrm e}^{\beta x}+\beta z}{\beta }\right ) {\mathrm e}^{\frac {c \,{\mathrm e}^{\gamma x}}{\gamma }}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },z-\frac {b e^{\beta x}}{\beta }\right ) \exp \left (\frac {c \text {Ei}\left (\frac {a e^{\lambda x} \gamma }{\lambda }\right ) e^{\gamma \left (y-\frac {a e^{\lambda x}}{\lambda }\right )}}{\lambda }+\frac {s \text {Ei}\left (\frac {b e^{\beta x} \mu }{\beta }\right ) e^{\mu \left (z-\frac {b e^{\beta x}}{\beta }\right )}}{\beta }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+\lambda y}{\lambda }, \frac {-b \,{\mathrm e}^{\beta x}+\beta z}{\beta }\right ) {\mathrm e}^{\frac {-\beta c \Ei \left (1, -\frac {a \gamma \,{\mathrm e}^{\lambda x}}{\lambda }\right ) {\mathrm e}^{-\frac {\left (a \,{\mathrm e}^{\lambda x}-\lambda y \right ) \gamma }{\lambda }}-\lambda s \Ei \left (1, -\frac {b \mu \,{\mathrm e}^{\beta x}}{\beta }\right ) {\mathrm e}^{\frac {\left (-b \,{\mathrm e}^{\beta x}+\beta z \right ) \mu }{\beta }}}{\beta \lambda }}\]
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-a \lambda x -{\mathrm e}^{-\lambda y}}{a \lambda }, \frac {-b \left ({\mathrm e}^{\lambda y}\right )^{\frac {\beta }{\lambda }} {\mathrm e}^{-\lambda y}+\left (\beta -\lambda \right ) a z}{\left (\beta -\lambda \right ) a}\right ) {\mathrm e}^{\int _{}^{x}\left (c \,{\mathrm e}^{\textit {\_a} \gamma }+s \,{\mathrm e}^{\frac {\left (-b \left ({\mathrm e}^{\lambda y}\right )^{\frac {\beta }{\lambda }} {\mathrm e}^{-\lambda y}+\left (\beta -\lambda \right ) a z +\left (\left (-\textit {\_a} +x \right ) a \lambda +{\mathrm e}^{-\lambda y}\right ) b \left (\frac {1}{\left (-\textit {\_a} +x \right ) a \lambda +{\mathrm e}^{-\lambda y}}\right )^{\frac {\beta }{\lambda }}\right ) \mu }{\left (\beta -\lambda \right ) a}}\right )d \textit {\_a}}\]
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\alpha 1 x}}{\alpha 1}+\nu 1 x}d x \right )-{\mathrm e}^{\frac {\left (\mathit {A1} \,{\mathrm e}^{\alpha 1 x}-\alpha 1 y \right ) \lambda }{\alpha 1}}}{\lambda }, \frac {-\mathit {B2} \beta \left (\int {\mathrm e}^{\frac {\mathit {A2} \beta \,{\mathrm e}^{\alpha 2 x}}{\alpha 2}+\nu 2 x}d x \right )-{\mathrm e}^{\frac {\left (\mathit {A2} \,{\mathrm e}^{\alpha 2 x}-\alpha 2 z \right ) \beta }{\alpha 2}}}{\beta }\right ) {\mathrm e}^{\frac {k \,{\mathrm e}^{\gamma x}}{\gamma }}\]
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (-\frac {\left (a \alpha \,{\mathrm e}^{\alpha x}-b \beta \,{\mathrm e}^{\beta y}\right ) {\mathrm e}^{-\alpha x -\beta y}}{\alpha b \beta }, -\frac {\left (a \alpha \,{\mathrm e}^{\alpha x}-c \gamma \,{\mathrm e}^{\gamma z}\right ) {\mathrm e}^{-\alpha x -\gamma z}}{\alpha c \gamma }\right ) {\mathrm e}^{-\frac {k \,{\mathrm e}^{-\left (\alpha -\lambda \right ) x}}{\left (\alpha -\lambda \right ) a}}\]
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \alpha \,{\mathrm e}^{\beta y}-b \beta \,{\mathrm e}^{\alpha x}}{\alpha b \beta }, -\frac {\left (a \alpha \,{\mathrm e}^{\beta y} {\mathrm e}^{-\gamma z}+\alpha c \gamma x -b \beta \,{\mathrm e}^{\alpha x} {\mathrm e}^{-\gamma z}-c \gamma \ln \left (\frac {a \alpha \,{\mathrm e}^{\beta y}}{b \beta }\right )\right ) b \beta }{\left (a \alpha \,{\mathrm e}^{\beta y}-b \beta \,{\mathrm e}^{\alpha x}\right ) \alpha c \gamma }\right ) {\mathrm e}^{\int _{}^{x}\frac {\alpha k \,{\mathrm e}^{\textit {\_a} \lambda }}{a \alpha \,{\mathrm e}^{\beta y}-\left (-{\mathrm e}^{\textit {\_a} \alpha }+{\mathrm e}^{\alpha x}\right ) b \beta }d \textit {\_a}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {k1} x}{\text {a1}}} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {a1} \text {k2}-\text {a2} \text {k1}}{\text {a1} \text {a2} \alpha }} c_1\left (\frac {\log \left (\frac {e^{\beta y} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {b1} \beta }{\text {a1} \alpha }}}{\text {b1}+\text {b2} e^{\beta y}}\right )}{\text {b1} \beta }-\frac {x}{\text {a1}},\frac {\log \left (\frac {e^{\gamma z} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {c1} \gamma }{\text {a1} \alpha }}}{\text {c1}+\text {c2} e^{\gamma z}}\right )}{\text {c1} \gamma }-\frac {x}{\text {a1}}\right )\right \}\right \}\]
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 (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {k1}}{\mathit {a1} \alpha }+\frac {\mathit {k2}}{\mathit {a2} \alpha }} \left ({\mathrm e}^{\alpha x}\right )^{\frac {\mathit {k1}}{\mathit {a1} \alpha }} \textit {\_F1} \left (\frac {-\mathit {b1} \beta x +\mathit {a1} \ln \left (\frac {\left (-\mathit {b1} +\RootOf \left (\mathit {a1} \alpha \beta y -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {b1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {b1} \beta }{\mathit {a1} \alpha }}}{\mathit {b2}}\right )+\mathit {b1} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )\right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {b1} \beta }{\mathit {a1} \alpha }}}{\mathit {b2} \RootOf \left (\mathit {a1} \alpha \beta y -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {b1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {b1} \beta }{\mathit {a1} \alpha }}}{\mathit {b2}}\right )+\mathit {b1} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}\right )}{\mathit {a1} \mathit {b1} \beta }, \frac {-\mathit {c1} \gamma x +\mathit {a1} \ln \left (\frac {\left (-\mathit {c1} +\RootOf \left (\mathit {a1} \alpha \gamma z -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {c1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {c1} \gamma }{\mathit {a1} \alpha }}}{\mathit {c2}}\right )+\mathit {c1} \gamma \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )\right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {c1} \gamma }{\mathit {a1} \alpha }}}{\mathit {c2} \RootOf \left (\mathit {a1} \alpha \gamma z -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {c1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {c1} \gamma }{\mathit {a1} \alpha }}}{\mathit {c2}}\right )+\mathit {c1} \gamma \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}\right )}{\mathit {a1} \mathit {c1} \gamma }\right )\]
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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]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {\text {k3} \left ((\text {a1} \text {k2}-\text {a2} \text {k1}) \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )+\text {a2} \alpha \text {k1} x\right )}{\text {a1} \text {a2} \alpha }\right ) c_1\left (\frac {c \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a1} \alpha }-\frac {c x}{\text {a1}}-\frac {e^{-\gamma z}}{\gamma },\frac {\log \left (\text {b1}+\text {b2} e^{\beta y}\right )}{\text {b2} \beta }-\frac {\log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a2} \alpha }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {k1} \mathit {k3}}{\mathit {a1} \alpha }+\frac {\mathit {k2} \mathit {k3}}{\mathit {a2} \alpha }} \left ({\mathrm e}^{\alpha x}\right )^{\frac {\mathit {k1} \mathit {k3}}{\mathit {a1} \alpha }} \textit {\_F1} \left (\frac {\ln \left (\frac {\mathit {b1} \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {b2} \beta }{\mathit {a2} \alpha }} \RootOf \left (\mathit {a2} \alpha \beta y -\mathit {a2} \alpha \ln \left (\frac {\mathit {b1} \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {b2} \beta }{\mathit {a2} \alpha }}}{\textit {\_Z} -\mathit {b2}}\right )-\mathit {b2} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}{-\mathit {b2} +\RootOf \left (\mathit {a2} \alpha \beta y -\mathit {a2} \alpha \ln \left (\frac {\mathit {b1} \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {b2} \beta }{\mathit {a2} \alpha }}}{\textit {\_Z} -\mathit {b2}}\right )-\mathit {b2} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}\right )}{\mathit {b2} \beta }, \frac {c \gamma \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )-\left (c \gamma x +\mathit {a1} \,{\mathrm e}^{-\gamma z}\right ) \alpha }{\mathit {a1} \alpha c \gamma }\right )\]
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