Added May 18, 2019.
Problem Chapter 6.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x+ b e^{\alpha x} w_y +c e^{\beta y} w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] +b*Exp[alpha*x]*D[w[x, y,z], y] +c*Exp[beta*y]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b e^{\alpha x}}{a \alpha },z-\frac {c \text {Ei}\left (\frac {b \beta e^{\alpha x}}{a \alpha }\right ) e^{\beta \left (y-\frac {b e^{\alpha x}}{a \alpha }\right )}}{a \alpha }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+b*exp(alpha*x)*diff(w(x,y,z),y)+c*exp(beta*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya\alpha -b{{\rm e}^{\alpha \,x}}}{a\alpha }},{\frac {1}{a\alpha } \left ( za\alpha +c{{\rm e}^{{\frac {\beta \, \left ( ya\alpha -b{{\rm e}^{\alpha \,x}} \right ) }{a\alpha }}}}\Ei \left ( 1,-{\frac {\beta \,b{{\rm e}^{\alpha \,x}}}{a\alpha }} \right ) \right ) } \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x+ b e^{\alpha x} w_y +c e^{\gamma z} w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}-\frac {e^{-\gamma z}}{\gamma },y-\frac {b e^{\alpha x}}{a \alpha }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya\alpha -b{{\rm e}^{\alpha \,x}}}{a\alpha }},{\frac {-c\gamma \,x-a{{\rm e}^{-\gamma \,z}}}{\gamma \,c}} \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x+ b e^{\beta y} w_y +c e^{\gamma z} w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {e^{-\beta y}}{\beta },-\frac {c x}{a}-\frac {e^{-\gamma z}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-b\beta \,x-a{{\rm e}^{-\beta \,y}}}{\beta \,b}},{\frac {-c\gamma \,x-a{{\rm e}^{-\gamma \,z}}}{c\gamma }} \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+(A_1 e^{\alpha _1 x} + B_1 e^{\nu _1 x+\lambda y}) w_y + (A_2 e^{\alpha _2 x} + B_2 e^{\nu _2 x+\beta y}) w_z = 0 \]
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*y])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; 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*y))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\lambda } \left ( -{\it B1}\,\int \!{{\rm e}^{\nu 1\,x+{\frac {\lambda \,{\it A1}\,{{\rm e}^{\alpha 1\,x}}}{\alpha 1}}}}\,{\rm d}x\lambda -{{\rm e}^{{\frac {\lambda \, \left ( {\it A1}\,{{\rm e}^{\alpha 1\,x}}-\alpha 1\,y \right ) }{\alpha 1}}}} \right ) },-\int ^{x}\!{\it A2}\,{{\rm e}^{\alpha 2\,{\it \_b}}}+ \left ( {\it B1}\,\int \!{{\rm e}^{\nu 1\,x+{\frac {\lambda \,{\it A1}\,{{\rm e}^{\alpha 1\,x}}}{\alpha 1}}}}\,{\rm d}x\lambda -{\it B1}\,\int \!{{\rm e}^{{\frac {\lambda \,{\it A1}\,{{\rm e}^{\alpha 1\,{\it \_b}}}+\nu 1\,{\it \_b}\,\alpha 1}{\alpha 1}}}}\,{\rm d}{\it \_b}\lambda +{{\rm e}^{{\frac {\lambda \, \left ( {\it A1}\,{{\rm e}^{\alpha 1\,x}}-\alpha 1\,y \right ) }{\alpha 1}}}} \right ) ^{-{\frac {\beta }{\lambda }}}{\it B2}\,{{\rm e}^{{\frac {\beta \,{\it A1}\,{{\rm e}^{\alpha 1\,{\it \_b}}}+\nu 2\,{\it \_b}\,\alpha 1}{\alpha 1}}}}{d{\it \_b}}+z \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a e^{\alpha x}w_x+ b e^{\beta y} w_y + c e^{\gamma z} w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b e^{-\alpha x}}{a \alpha }-\frac {e^{-\beta y}}{\beta },\frac {c e^{-\alpha x}}{a \alpha }-\frac {e^{-\gamma z}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}\beta \,b-a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }},{\frac { \left ( {{\rm e}^{\gamma \,z}}\gamma \,c-a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\gamma \,z}}}{\gamma \,\alpha \,c}} \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a e^{\beta y}w_x+ b e^{\alpha x} w_y + c e^{\gamma z} w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{\beta y}}{\beta }-\frac {b e^{\alpha x}}{a \alpha },-\frac {c \gamma \log \left (\frac {a \alpha e^{\beta y}}{\beta }\right )-a \alpha e^{\beta y-\gamma z}+b \beta e^{\alpha x-\gamma z}-\alpha c \gamma x}{b \beta \gamma e^{\alpha x}-a \alpha \gamma e^{\beta y}}\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{{\rm e}^{\beta \,y}}a\alpha -{{\rm e}^{\alpha \,x}}b\beta }{\alpha \,b\beta }},-{\frac {\beta \,b}{\gamma \, \left ( {{\rm e}^{\beta \,y}}a\alpha -{{\rm e}^{\alpha \,x}}b\beta \right ) \alpha \,c} \left ( a{{\rm e}^{\beta \,y}}{{\rm e}^{-\gamma \,z}}\alpha -{{\rm e}^{\alpha \,x}}{{\rm e}^{-\gamma \,z}}b\beta +\gamma \,\alpha \,cx-\ln \left ( {\frac {{{\rm e}^{\beta \,y}}a\alpha }{\beta \,b}} \right ) \gamma \,c \right ) } \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (a_1+ a_2 e^{\alpha x}) w_x+ (b_1 + b_2 e^{\beta y} w_y + (c_1+c_2 e^{\gamma z}) w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\frac {e^{\beta y} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {b1} \beta }{\text {a1} \alpha }}}{\text {b1}+\text {b2} e^{\beta y}}\right )}{\text {b1} \beta }-\frac {x}{\text {a1}},\frac {\log \left (\frac {e^{\gamma z} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {c1} \gamma }{\text {a1} \alpha }}}{\text {c1}+\text {c2} e^{\gamma z}}\right )}{\text {c1} \gamma }-\frac {x}{\text {a1}}\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\alpha \,{\it a1}\,\beta \,{\it b1}} \left ( -\alpha \,{\it a1}\,\RootOf \left ( y\alpha \,{\it a1}\,\beta -\alpha \,{\it a1}\,\ln \left ( {\frac {-{\it b1}+{{\rm e}^{{\it \_Z}}}}{{\it b2}} \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{{\frac {\beta \,{\it b1}}{\alpha \,{\it a1}}}}} \right ) +\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \beta \,{\it b1} \right ) - \left ( -\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) {\it b1}+\alpha \, \left ( -{\it a1}\,y+{\it b1}\,x \right ) \right ) \beta \right ) },{\frac {1}{\alpha \,{\it a1}\,\gamma \,{\it c1}} \left ( -\alpha \,{\it a1}\,\RootOf \left ( z\alpha \,{\it a1}\,\gamma +\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \gamma \,{\it c1}-\alpha \,{\it a1}\,\ln \left ( {\frac {-{\it c1}+{{\rm e}^{{\it \_Z}}}}{{\it c2}} \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{{\frac {\gamma \,{\it c1}}{\alpha \,{\it a1}}}}} \right ) \right ) - \left ( -{\it c1}\,\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) +\alpha \, \left ( -{\it a1}\,z+{\it c1}\,x \right ) \right ) \gamma \right ) } \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ e^{\beta y} (a_1+ a_2 e^{\alpha x}) w_x+ e^{\alpha x}(b_1 + b_2 e^{\beta y} w_y +c e^{\beta y + \gamma z} w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {c \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a1} \alpha }-\frac {c x}{\text {a1}}-\frac {e^{-\gamma z}}{\gamma },\frac {\log \left (\text {b1}+\text {b2} e^{\beta y}\right )}{\text {b2} \beta }-\frac {\log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a2} \alpha }\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\alpha \,{\it a2}\,\beta \,{\it b2}} \left ( y\alpha \,{\it a2}\,\beta -\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \beta \,{\it b2}+\alpha \,{\it a2}\,\RootOf \left ( y\alpha \,{\it a2}\,\beta -\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \beta \,{\it b2}-\alpha \,{\it a2}\,\ln \left ( {\frac {{\it b1}}{-{\it b2}+{{\rm e}^{{\it \_Z}}}} \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{-{\frac {\beta \,{\it b2}}{\alpha \,{\it a2}}}}} \right ) \right ) \right ) },{\frac {\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \gamma \,c-\alpha \, \left ( c\gamma \,x+{\it a1}\,{{\rm e}^{-\gamma \,z}} \right ) }{\gamma \,{\it a1}\,\alpha \,c}} \right ) \]
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