Added May 18, 2019.
Problem Chapter 6.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a y e^{\alpha x} w_x+ b e^{\beta y} w_y +c e^{\gamma z} w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*y*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]];
Failed
Maple ✓
restart; pde := a*y*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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac { \left ( a\alpha \,{{\rm e}^{\alpha \,x-\beta \,y}} \left ( \beta \,y+1 \right ) -b{\beta }^{2} \right ) {{\rm e}^{-\alpha \,x}}}{\alpha \,b{\beta }^{2}}},{\frac {{{\rm e}^{-\gamma \,z-1}}}{b{\beta }^{2}{\alpha }^{2}\gamma \,ac} \left ( \gamma \,{{\rm e}^{\gamma \,z}}\int ^{x}\!-{\frac {{{\rm e}^{-\alpha \,{\it \_a}+1}}b{\beta }^{2}}{a{{\rm e}^{-\alpha \, \left ( x-{\it \_a} \right ) }}\alpha \, \left ( \beta \,y+1 \right ) {{\rm e}^{\alpha \,x-\beta \,y}}-b{\beta }^{2} \left ( {{\rm e}^{-\alpha \, \left ( x-{\it \_a} \right ) }}-1 \right ) } \left ( \LambertW \left ( -{\frac { \left ( a{{\rm e}^{-\alpha \, \left ( x-{\it \_a} \right ) }}\alpha \, \left ( \beta \,y+1 \right ) {{\rm e}^{\alpha \,x-\beta \,y}}-b{\beta }^{2} \left ( {{\rm e}^{-\alpha \, \left ( x-{\it \_a} \right ) }}-1 \right ) \right ) {{\rm e}^{-\alpha \,{\it \_a}-1}}}{a\alpha }} \right ) +1 \right ) ^{-1}}{d{\it \_a}}{\beta }^{2}cb\alpha + \left ( a\alpha \,{{\rm e}^{\alpha \,x-\beta \,y}} \left ( \beta \,y+1 \right ) -b{\beta }^{2} \right ) \gamma \,c{{\rm e}^{-\alpha \,x+\gamma \,z+1}}\ln \left ( \LambertW \left ( -{{\rm e}^{\alpha \,x-\beta \,y}} \left ( \beta \,y+1 \right ) {{\rm e}^{-\alpha \,x-1}} \right ) \right ) +a{\rm e}b\beta \,\alpha \right ) } \right ) \]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a y e^{\alpha x} w_x+ b e^{\beta y} w_y +c z e^{\gamma z} w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*y*Exp[alpha*x]*D[w[x, y,z], x] + b*Exp[beta*y]*D[w[x, y,z], y] +c*z*Exp[gamma*z]*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 := a*y*exp(alpha*x)*diff(w(x,y,z),x)+b*exp(beta*y)*diff(w(x,y,z),y)+c*z*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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac { \left ( a\alpha \,{{\rm e}^{\alpha \,x-\beta \,y}} \left ( \beta \,y+1 \right ) -b{\beta }^{2} \right ) {{\rm e}^{-\alpha \,x}}}{\alpha \,b{\beta }^{2}}},{\frac {\Ei \left ( 1,\gamma \,z \right ) b\beta \,\LambertW \left ( -{{\rm e}^{\alpha \,x-\beta \,y}} \left ( \beta \,y+1 \right ) {{\rm e}^{-\alpha \,x-1}} \right ) +{{\rm e}^{-\alpha \,x}}{{\rm e}^{\alpha \,x-\beta \,y}} \left ( \beta \,y+1 \right ) c}{\alpha \,b{\beta }^{2}\LambertW \left ( -{{\rm e}^{\alpha \,x-\beta \,y}} \left ( \beta \,y+1 \right ) {{\rm e}^{-\alpha \,x-1}} \right ) c}} \right ) \]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left [ y^2+ a e^{\alpha x}(\alpha -a e^{\alpha x}) \right ] w_y +\left [ z^2 +b z +c e^{\beta x}(\beta - b -c e^{\beta x}) \right ] w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( y^2+ a*Exp[alpha*x]*(alpha-a*Exp[alpha*x]))*D[w[x, y,z], y] +(z^2 +b*z +c*Exp[beta*x]*(beta - b -c*Exp[beta*x]))*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\text {Ei}\left (\frac {2 a e^{\alpha x}}{\alpha }\right ) \left (y-a e^{\alpha x}\right )+\alpha e^{\frac {2 a e^{\alpha x}}{\alpha }}}{a e^{\alpha x}-y},\frac {2^{b/\beta } \beta ^{-\frac {b}{\beta }} e^{b x} c^{b/\beta } \left (\left (b-c e^{\beta x}+z\right ) \text {LaguerreL}\left (-\frac {b}{\beta },\frac {b}{\beta },\frac {2 c e^{\beta x}}{\beta }\right )-2 c e^{\beta x} \text {LaguerreL}\left (-\frac {b+\beta }{\beta },\frac {b+\beta }{\beta },\frac {2 c e^{\beta x}}{\beta }\right )\right )}{c e^{\beta x}-z}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+( y^2+ a*exp(alpha*x)*(alpha-a*exp(alpha*x)))*diff(w(x,y,z),y)+(z^2 +b*z +c*exp(beta*x)*(beta - b -c*exp(beta*x)))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {(-a{{\rm e}^{\alpha \,x}}+y) \left ( \left ( a{{\rm e}^{\alpha \,x}}-y \right ) \Ei \left ( 1,-2\,{\frac {a{{\rm e}^{\alpha \,x}}}{\alpha }} \right ) +{{\rm e}^{2\,{\frac {a{{\rm e}^{\alpha \,x}}}{\alpha }}}}\alpha \right ) ^{-1}},{\frac {1}{c{{\rm e}^{\beta \,x}}-z} \left ( \left ( -c{{\rm e}^{\beta \,x}}+z \right ) \int \!{{\rm e}^{{\frac {b\beta \,x+2\,c{{\rm e}^{\beta \,x}}}{\beta }}}}\,{\rm d}x+{{\rm e}^{{\frac {b\beta \,x+2\,c{{\rm e}^{\beta \,x}}}{\beta }}}} \right ) } \right ) \]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left [ y^2+ b y + a e^{\alpha x}(\alpha -b - a e^{\alpha x}) \right ] w_y +\left [ z^2 +c e^{\beta x}(z-k)-k^2 \right ] w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( y^2+ b*y + a*Exp[alpha*x]*(alpha-b-a*Exp[alpha*x]))*D[w[x, y,z], y] +(z^2 +c*Exp[beta*x]*(z-k)-k^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (2 k (-1)^{-\frac {k}{\beta }} \left (-\frac {\text {Gamma}\left (\frac {2 k}{\beta },0,-\frac {c e^{\beta x}}{\beta }\right )}{\beta }+\frac {\beta ^{-\frac {2 k}{\beta }} c^{\frac {2 k}{\beta }} e^{\frac {c e^{\beta x}+2 \beta k x+2 i \pi k}{\beta }}}{k-z}\right ),\frac {2^{\frac {b}{\alpha }} \alpha ^{-\frac {b}{\alpha }} e^{b x} a^{\frac {b}{\alpha }} \left (\left (a \left (-e^{\alpha x}\right )+b+y\right ) \text {LaguerreL}\left (-\frac {b}{\alpha },\frac {b}{\alpha },\frac {2 a e^{\alpha x}}{\alpha }\right )-2 a e^{\alpha x} \text {LaguerreL}\left (-\frac {\alpha +b}{\alpha },\frac {\alpha +b}{\alpha },\frac {2 a e^{\alpha x}}{\alpha }\right )\right )}{a e^{\alpha x}-y}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+( y^2+ b*y+a*exp(alpha*x)*(alpha-b-a*exp(alpha*x)))*diff(w(x,y,z),y)+(z^2 +c*exp(beta*x)*(z-k)-k^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{a{{\rm e}^{\alpha \,x}}-y} \left ( \left ( -a{{\rm e}^{\alpha \,x}}+y \right ) \int \!{{\rm e}^{{\frac {\alpha \,xb+2\,a{{\rm e}^{\alpha \,x}}}{\alpha }}}}\,{\rm d}x+{{\rm e}^{{\frac {\alpha \,xb+2\,a{{\rm e}^{\alpha \,x}}}{\alpha }}}} \right ) },{\frac {1}{-z+k} \left ( \left ( -z+k \right ) \int \!{{\rm e}^{{\frac {2\,kx\beta +c{{\rm e}^{\beta \,x}}}{\beta }}}}\,{\rm d}x-{{\rm e}^{{\frac {2\,kx\beta +c{{\rm e}^{\beta \,x}}}{\beta }}}} \right ) } \right ) \]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left [ y^2+ b y + a e^{\alpha x}(y-b)-b^2 \right ] w_y +\left [ z^2 +c(x z-1)e^{\beta x} \right ] w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( y^2+ b*y + a*Exp[alpha*x]*(y-b)-b^2)*D[w[x, y,z], y] +(z^2 +c*(x*z-1)*Exp[beta*x])*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)+( y^2+ b*y+a*exp(alpha*x)*(y-b)-b^2)*diff(w(x,y,z),y)+(z^2 +c*(x*z-1)*exp(beta*x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
server hangs
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Added May 18, 2019.
Problem Chapter 6.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left [ y^2 - a e^{\alpha x}(x y-1) \right ] w_y +\left (c e^{\beta x} z^2+ b e^{-\beta x} \right ) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( y^2- a*Exp[alpha*x]*(x*y-1))*D[w[x, y,z], y] +(c*Exp[beta*x]*z^2+b*Exp[-beta*x])*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)+( y^2- a*exp(alpha*x)*(x*y-1))*diff(w(x,y,z),y)+(c*exp(beta*x)*z^2+b*exp(-beta*x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
server hangs
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left ( a y^2 e^{\alpha x} + b e^{-\alpha x} \right ) w_y +\left [ d e^{\beta x} z^2+ c e^{\gamma x}(\gamma - c d e^{(\beta +\gamma )x} ) \right ] w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( a*y^2- a*Exp[alpha*x]+ b * Exp[-alpha*x])*D[w[x, y,z], y] +(d*Exp[beta*x]*z^2+c*Exp[gamma*x]*(gamma- c*d*Exp[(beta+gamma)*x]))*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)+( a*y^2- a*exp(alpha*x)+ b * exp(-alpha*x))*diff(w(x,y,z),y)+(d*exp(beta*x)*z^2+c*exp(gamma*x)*(gamma- c*d*exp((beta+gamma)*x)))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
server hangs
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left [ b e^{\alpha x} y^2 + a e^{\beta x} (\beta - a b e^{(\alpha +\beta )x}) \right ] w_y +\left ( c z^2 e^{\gamma x}+ d z + k e^{-\gamma x} \right ) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (b*Exp[alpha*x]*y^2 + a*Exp[beta*x]*(beta- a*b*Exp[(alpha+beta)*x]))*D[w[x, y,z], y] +(c*z^2*Exp[gamma*x]+ d*z + k*Exp[-gamma*x])*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)+ (b*exp(alpha*x)*y^2 + a*exp(beta*x)*(beta- a*b*exp((alpha+beta)*x)))*diff(w(x,y,z),y)+(c*z^2*exp(gamma*x)+ d*z + k*exp(-gamma*x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
server hangs
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left ( a e^{\alpha x} y^2 + b y + c e^{\alpha x} \right ) w_y +\left (e^{\beta x} z^2+ d e^{\gamma x}(z+\beta e^{-\beta x}) \right ) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*Exp[alpha*x]*y^2 + b*y + c*Exp[alpha*x])*D[w[x, y,z], y] +(Exp[beta*x]*z^2+ d*Exp[gamma*x]*(z+beta*Exp[-beta*x]))*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)+ (a*exp(alpha*x)*y^2 + b*y + c*exp(alpha*x))*diff(w(x,y,z),y)+(exp(beta*x)*z^2+ d*exp(gamma*x)*(z+beta*exp(-beta*x)))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( { \left ( -y\sqrt {a}\BesselJ \left ( -{\frac {\alpha +b}{2\,\alpha }},{\frac {{{\rm e}^{\alpha \,x}}}{\alpha }\sqrt {a}\sqrt {c}} \right ) +\sqrt {c}\BesselJ \left ( {\frac {\alpha -b}{2\,\alpha }},{\frac {{{\rm e}^{\alpha \,x}}}{\alpha }\sqrt {a}\sqrt {c}} \right ) \right ) \left ( y\sqrt {a}\BesselY \left ( -{\frac {\alpha +b}{2\,\alpha }},{\frac {{{\rm e}^{\alpha \,x}}}{\alpha }\sqrt {a}\sqrt {c}} \right ) -\BesselY \left ( {\frac {\alpha -b}{2\,\alpha }},{\frac {{{\rm e}^{\alpha \,x}}}{\alpha }\sqrt {a}\sqrt {c}} \right ) \sqrt {c} \right ) ^{-1}},-{ \left ( -\gamma +\beta \right ) {{\rm e}^{\beta \,x}} \left ( {{\rm e}^{\beta \,x}}z+\beta \right ) \left ( \hypergeom \left ( [{\frac {\gamma -\beta }{\gamma }}],[{\frac {2\,\gamma -\beta }{\gamma }}],{\frac {{{\rm e}^{\gamma \,x}}d}{\gamma }} \right ) {{\rm e}^{\gamma \,x}}\beta \,d+z{{\rm e}^{\beta \,x}}\hypergeom \left ( [-{\frac {\beta }{\gamma }}],[{\frac {\gamma -\beta }{\gamma }}],{\frac {{{\rm e}^{\gamma \,x}}d}{\gamma }} \right ) \left ( -\gamma +\beta \right ) \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left [ e^{\alpha x} y^2 + a y e^{\beta x} + a \alpha e^{(\beta -\alpha )x} \right ] w_y +\left [ \gamma e^{\gamma x} z^2+ b e^{\delta x}(z+e^{-\gamma x}) \right ] w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( Exp[alpha*x]*y^2 + a*y*Exp[beta*x] + a*alpha*Exp[(beta-alpha)*x])*D[w[x, y,z], y] +(gamma*Exp[gamma*x]*z^2+ b*Exp[delta*x]*(z+Exp[-gamma*x]))*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)+ ( exp(alpha*x)*y^2 + a*y*exp(beta*x) + a*alpha*exp((beta-alpha)*x))*diff(w(x,y,z),y)+(gamma*exp(gamma*x)*z^2+ b*exp(delta*x)*(z+exp(-gamma*x)))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {{{\rm e}^{\alpha \,x}} \left ( \beta -\alpha \right ) \left ( {{\rm e}^{\alpha \,x}}y+\alpha \right ) \left ( {{\rm e}^{\beta \,x}}\hypergeom \left ( [{\frac {\beta -\alpha }{\beta }}],[{\frac {-\alpha +2\,\beta }{\beta }}],{\frac {a{{\rm e}^{\beta \,x}}}{\beta }} \right ) a\alpha -y{{\rm e}^{\alpha \,x}}\hypergeom \left ( [-{\frac {\alpha }{\beta }}],[{\frac {\beta -\alpha }{\beta }}],{\frac {a{{\rm e}^{\beta \,x}}}{\beta }} \right ) \left ( \beta -\alpha \right ) \right ) ^{-1}},{{{\rm e}^{\gamma \,x}} \left ( -\gamma +\delta \right ) \left ( z{{\rm e}^{\gamma \,x}}+1 \right ) \left ( \hypergeom \left ( [{\frac {-\gamma +\delta }{\delta }}],[{\frac {-\gamma +2\,\delta }{\delta }}],{\frac {b{{\rm e}^{\delta \,x}}}{\delta }} \right ) b{{\rm e}^{\delta \,x}}-z{{\rm e}^{\gamma \,x}}\hypergeom \left ( [-{\frac {\gamma }{\delta }}],[{\frac {-\gamma +\delta }{\delta }}],{\frac {b{{\rm e}^{\delta \,x}}}{\delta }} \right ) \left ( -\gamma +\delta \right ) \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+\left [ \alpha e^{\alpha x} y^2 + a e^{\beta x} (y+e^{-\alpha x}) \right ] w_y +\left [ e^{\gamma x} (z- b e^{\delta x})^2 + b \delta e^{\delta x} \right ] w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( alpha*Exp[alpha*x]*y^2 + a*Exp[beta*x]*(y+Exp[-alpha*x]))*D[w[x, y,z], y] +(Exp[gamma*x]*(z-b*Exp[delta*x])^2+b*delta*Exp[delta*x])*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)+ ( alpha*exp(alpha*x)*y^2 + a*exp(beta*x)*(y+exp(-alpha*x)))*diff(w(x,y,z),y)+(exp(gamma*x)*(z-b*exp(delta*x))^2+b*delta*exp(delta*x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {{{\rm e}^{\alpha \,x}} \left ( \beta -\alpha \right ) \left ( {{\rm e}^{\alpha \,x}}y+1 \right ) \left ( {{\rm e}^{\beta \,x}}\hypergeom \left ( [{\frac {\beta -\alpha }{\beta }}],[{\frac {-\alpha +2\,\beta }{\beta }}],{\frac {a{{\rm e}^{\beta \,x}}}{\beta }} \right ) a-y{{\rm e}^{\alpha \,x}}\hypergeom \left ( [-{\frac {\alpha }{\beta }}],[{\frac {\beta -\alpha }{\beta }}],{\frac {a{{\rm e}^{\beta \,x}}}{\beta }} \right ) \left ( \beta -\alpha \right ) \right ) ^{-1}},{\frac {b{{\rm e}^{\delta \,x}}-z}{\gamma \, \left ( -b{{\rm e}^{x \left ( \gamma +\delta \right ) }}+z{{\rm e}^{\gamma \,x}}+\gamma \right ) }} \right ) \]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x+\left ( a_1 e^{\alpha x} y^2 + \beta y+ a_1 b_2^2 x^{2 \beta } e^{\alpha x} \right ) w_y +\left [ a_2 x^{2 n} z^2 e^{\lambda x}+(b_2 x^n e^{\lambda x} - n) z + c e^{\lambda x} \right ] w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + ( a1*Exp[alpha*x]*y^2 + beta*y+ a1*b2^2*x^(2*beta)*Exp[alpha*x])*D[w[x, y,z], y] +(a2*x^(2*n)*z^2*Exp[lamba*x]+(b2*x^n*Exp[lambda*x] - n)*z + c*Exp[lambda*x])*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 := x*diff(w(x,y,z),x)+ ( a1*exp(alpha*x)*y^2 + beta*y+ a1*b2^2*x^(2*beta)*exp(alpha*x))*diff(w(x,y,z),y)+(a2*x^(2*n)*z^2*exp(lamba*x)+(b2*x^n*exp(lambda*x) - n)*z + c*exp(lambda*x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
server hangs
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.3.2.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( a_1 e^{\lambda _1 x} y + b_1 e^{\beta _1 x} y^k \right ) w_y + \left ( a_2 e^{\lambda _2 x} z + b_2 e^{\beta _1 x} z^m \right ) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +( a1*Exp[lambda1*x]*y + b1*Exp[beta1*x]*y^k)*D[w[x, y,z], y] +(a2*Exp[lamba2*x]*z + b2*Exp[beta1*x]*z^m) *D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left ((k-1) \int _1^x\text {b1} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {beta1} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {a2} e^{\text {lamba2} K[2]} (m-1)}{\text {lamba2}}+\text {beta1} K[2]}dK[2]+z^{1-m} e^{\frac {\text {a2} (m-1) e^{\text {lamba2} x}}{\text {lamba2}}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (a1*exp(lambda1*x)*y + b1*exp(beta1*x)*y^k)*diff(w(x,y,z),y)+(a2*exp(lamba2*x)*z + b2*exp(beta1*x)*z^m)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}}{\lambda 1}}}}{\it b1}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}} \left ( k-1 \right ) +\beta 1\,x\lambda 1}{\lambda 1}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}k}{\lambda 1}}}}y \right ) \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}}{\lambda 1}}}} \right ) ^{-1}},{\frac {1}{{z}^{m}} \left ( {{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{{\it lamba2}\,x}}}{{\it lamba2}}}}}{\it b2}\,{z}^{m} \left ( m-1 \right ) \int \!{{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{{\it lamba2}\,x}} \left ( m-1 \right ) +\beta 1\,x{\it lamba2}}{{\it lamba2}}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{{\it lamba2}\,x}}m}{{\it lamba2}}}}}z \right ) \left ( {{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{{\it lamba2}\,x}}}{{\it lamba2}}}}} \right ) ^{-1}} \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.2.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( a_1 e^{\beta _1 x} y + b_1 e^{\gamma _1 x} y^k \right ) w_y + \left ( a_2 e^{\beta _2 x} + b_2 e^{\gamma _1 x+\lambda z} \right ) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +( a1*Exp[beta1*x]*y + b1*Exp[gamma1*x]*y^k)*D[w[x, y,z], y] +(a2*Exp[beta2*x] + b2*Exp[gamma1*x+lambda*z]) *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(beta1*x)*y + b1*exp(gamma1*x)*y^k)*diff(w(x,y,z),y)+(a2*exp(beta2*x)+ b2*exp(gamma1*x+lambda*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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}}}{\beta 1}}}}{\it b1}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}} \left ( k-1 \right ) +\gamma 1\,x\beta 1}{\beta 1}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}}k}{\beta 1}}}}y \right ) \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}}}{\beta 1}}}} \right ) ^{-1}},{\frac {1}{\lambda } \left ( -{\it b2}\,\int \!{{\rm e}^{\gamma 1\,x+{\frac {\lambda \,{\it a2}\,{{\rm e}^{\beta 2\,x}}}{\beta 2}}}}\,{\rm d}x\lambda -{{\rm e}^{{\frac {\lambda \, \left ( {\it a2}\,{{\rm e}^{\beta 2\,x}}-\beta 2\,z \right ) }{\beta 2}}}} \right ) } \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.2.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( a_1 x^n + b_1 x^m e^{\lambda y} \right ) w_y + \left ( a_2 x^k+b_2 x^s e^{\beta z} \right ) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +( a1*x^n+b1*x^m*Exp[lambda*y])*D[w[x, y,z], y] +(a2*x^k+b2*x^2*Exp[beta*z]) *D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\text {b2} \beta x^3 \left (-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )^{-\frac {3}{k+1}} \text {Gamma}\left (\frac {3}{k+1},-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\beta \left (-\text {a2} x^{k+1}+k z+z\right )}{k+1}}}{\text {a2} \text {b2} \beta ^2 \left (k^2-k-2\right )},\frac {(n+1) e^{-\frac {\lambda \left (-\text {a1} x^{n+1}+n y+y\right )}{n+1}}-\text {b1} \lambda x^{m+1} \left (-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \text {Gamma}\left (\frac {m+1}{n+1},-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )}{\text {a1} \text {b1} \lambda ^2 (n+1) (m-n)}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ ( a1*x^n+b1*x^m*exp(lambda*y))*diff(w(x,y,z),y)+(a2*x^k+b2*x^2*exp(beta*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}{ \left ( m+1 \right ) \left ( n+m+2 \right ) \left ( 2\,n+m+3 \right ) {\it a1}\,\lambda } \left ( {\it b1}\, \left ( -{\frac {{x}^{1+n}\lambda \,{\it a1}}{1+n}} \right ) ^{{\frac {-n-m-2}{2+2\,n}}} \left ( 1+n \right ) ^{2} \left ( \left ( n+m+2 \right ) {x}^{m-n}-{\it a1}\,\lambda \,{x}^{m+1} \right ) {{\rm e}^{{\frac {{x}^{1+n}\lambda \,{\it a1}}{2+2\,n}}}} \WhittakerM \left ( {\frac {m-n}{2+2\,n}},{\frac {2\,n+m+3}{2+2\,n}},-{\frac {{x}^{1+n}\lambda \,{\it a1}}{1+n}} \right ) - \left ( -{x}^{m-n}{{\rm e}^{{\frac {{x}^{1+n}\lambda \,{\it a1}}{2+2\,n}}}} \left ( -{\frac {{x}^{1+n}\lambda \,{\it a1}}{1+n}} \right ) ^{{\frac {-n-m-2}{2+2\,n}}}{\it b1}\, \left ( 1+n \right ) \left ( n+m+2 \right ) \WhittakerM \left ( {\frac {n+m+2}{2+2\,n}},{\frac {2\,n+m+3}{2+2\,n}},-{\frac {{x}^{1+n}\lambda \,{\it a1}}{1+n}} \right ) +{{\rm e}^{-{\frac {\lambda \, \left ( -{x}^{1+n}{\it a1}+y \left ( 1+n \right ) \right ) }{1+n}}}}{\it a1}\, \left ( m+1 \right ) \left ( 2\,n+m+3 \right ) \right ) \left ( n+m+2 \right ) \right ) },{\frac {1}{3\,{\it a2}\,\beta \, \left ( 2\,{k}^{2}+13\,k+20 \right ) } \left ( {{\rm e}^{{\frac {{x}^{1+k}\beta \,{\it a2}}{2\,k+2}}}} \left ( \left ( 4+k \right ) {x}^{-k+2}-{x}^{3}{\it a2}\,\beta \right ) {\it b2}\, \left ( 1+k \right ) ^{2} \left ( -{\frac {{x}^{1+k}\beta \,{\it a2}}{1+k}} \right ) ^{{\frac {-4-k}{2\,k+2}}} \WhittakerM \left ( {\frac {-k+2}{2\,k+2}},{\frac {5+2\,k}{2\,k+2}},-{\frac {{x}^{1+k}\beta \,{\it a2}}{1+k}} \right ) + \left ( 4+k \right ) \left ( \left ( -{\frac {{x}^{1+k}\beta \,{\it a2}}{1+k}} \right ) ^{{\frac {-4-k}{2\,k+2}}}{\it b2}\,{x}^{-k+2}{{\rm e}^{{\frac {{x}^{1+k}\beta \,{\it a2}}{2\,k+2}}}} \left ( 4+k \right ) \left ( 1+k \right ) \WhittakerM \left ( {\frac {4+k}{2\,k+2}},{\frac {5+2\,k}{2\,k+2}},-{\frac {{x}^{1+k}\beta \,{\it a2}}{1+k}} \right ) -6\,{\it a2}\, \left ( k+5/2 \right ) {{\rm e}^{-{\frac {\beta \, \left ( -{x}^{1+k}{\it a2}+z \left ( 1+k \right ) \right ) }{1+k}}}} \right ) \right ) } \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.2.16, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (a x^n e^{\lambda y} + b x y^m) w_x + e^{\mu y} w_y + \left ( c y^l z^k + d y^p z \right ) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*Exp[lambda*y] + b*x*y^m)*D[w[x, y,z], x] +Exp[mu*y]*D[w[x, y,z], y] +(c*y^L*z^k+d*y^p*z) *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 := (a*x^n*exp(lambda*y) + b*x*y^m)*diff(w(x,y,z),x)+ exp(mu*y)*diff(w(x,y,z),y)+(c*y^L*z^k+d*y^p*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 ( {{x}^{ \left ( m+1 \right ) ^{-1}}{{\rm e}^{{\frac {bn{y}^{m}}{\mu \, \left ( m+1 \right ) }{{\rm e}^{-{\frac {\mu \,y}{2}}}} \left ( \mu \,y \right ) ^{-{\frac {m}{2}}} \WhittakerM \left ( {\frac {m}{2}},{\frac {1}{2}}+{\frac {m}{2}},\mu \,y \right ) }}}{x}^{{\frac {m}{m+1}}} \left ( {x}^{{\frac {mn}{m+1}}} \right ) ^{-1} \left ( {x}^{{\frac {n}{m+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{y}^{m}}{\mu \, \left ( m+1 \right ) }{{\rm e}^{-{\frac {\mu \,y}{2}}}} \left ( \mu \,y \right ) ^{-{\frac {m}{2}}} \WhittakerM \left ( {\frac {m}{2}},{\frac {1}{2}}+{\frac {m}{2}},\mu \,y \right ) }}} \right ) ^{-1}}+na\int \!{{\rm e}^{{\frac {1}{\mu \, \left ( m+1 \right ) } \left ( b{y}^{m} \left ( \mu \,y \right ) ^{-{\frac {m}{2}}}{{\rm e}^{-{\frac {\mu \,y}{2}}}} \left ( n-1 \right ) \WhittakerM \left ( {\frac {m}{2}},{\frac {1}{2}}+{\frac {m}{2}},\mu \,y \right ) -\mu \,y \left ( \mu -\lambda \right ) \left ( m+1 \right ) \right ) }}}\,{\rm d}y-a\int \!{{\rm e}^{{\frac {1}{\mu \, \left ( m+1 \right ) } \left ( b{y}^{m} \left ( \mu \,y \right ) ^{-{\frac {m}{2}}}{{\rm e}^{-{\frac {\mu \,y}{2}}}} \left ( n-1 \right ) \WhittakerM \left ( {\frac {m}{2}},{\frac {1}{2}}+{\frac {m}{2}},\mu \,y \right ) -\mu \,y \left ( \mu -\lambda \right ) \left ( m+1 \right ) \right ) }}}\,{\rm d}y,{{z}^{ \left ( p+1 \right ) ^{-1}}{{\rm e}^{{\frac {dk{y}^{p}}{\mu \, \left ( p+1 \right ) } \left ( \mu \,y \right ) ^{-{\frac {p}{2}}}{{\rm e}^{-{\frac {\mu \,y}{2}}}} \WhittakerM \left ( {\frac {p}{2}},{\frac {p}{2}}+{\frac {1}{2}},\mu \,y \right ) }}}{z}^{{\frac {p}{p+1}}} \left ( {z}^{{\frac {pk}{p+1}}} \right ) ^{-1} \left ( {z}^{{\frac {k}{p+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {d{y}^{p}}{\mu \, \left ( p+1 \right ) } \left ( \mu \,y \right ) ^{-{\frac {p}{2}}}{{\rm e}^{-{\frac {\mu \,y}{2}}}} \WhittakerM \left ( {\frac {p}{2}},{\frac {p}{2}}+{\frac {1}{2}},\mu \,y \right ) }}} \right ) ^{-1}}+ck\int \!{{\rm e}^{{\frac {1}{\mu \, \left ( p+1 \right ) } \left ( d{y}^{p} \left ( \mu \,y \right ) ^{-{\frac {p}{2}}}{{\rm e}^{-{\frac {\mu \,y}{2}}}} \left ( k-1 \right ) \WhittakerM \left ( {\frac {p}{2}},{\frac {p}{2}}+{\frac {1}{2}},\mu \,y \right ) -{\mu }^{2} \left ( p+1 \right ) y \right ) }}}{y}^{L}\,{\rm d}y-c\int \!{{\rm e}^{{\frac {1}{\mu \, \left ( p+1 \right ) } \left ( d{y}^{p} \left ( \mu \,y \right ) ^{-{\frac {p}{2}}}{{\rm e}^{-{\frac {\mu \,y}{2}}}} \left ( k-1 \right ) \WhittakerM \left ( {\frac {p}{2}},{\frac {p}{2}}+{\frac {1}{2}},\mu \,y \right ) -{\mu }^{2} \left ( p+1 \right ) y \right ) }}}{y}^{L}\,{\rm d}y \right ) \]
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Added May 18, 2019.
Problem Chapter 6.3.2.17, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (y^2 + 2 a \alpha e^{\alpha x^2}-a^2 e^{2 \alpha x^2} )w_y + \left ( c e^{-2 \beta x^2} z^2 + 2 \beta x z + b^2 c \right ) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (y^2 + 2*a*alpha*Exp[alpha*x^2]-a^2*Exp[2*alpha*x^2] )*D[w[x, y,z], y] +( c*Exp[-2*beta*x^2]*z^2 + 2*beta*x*z + b^2*c)*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)+ (y^2 + 2*a*alpha*exp(alpha*x^2)-a^2*exp(2*alpha*x^2) )*diff(w(x,y,z),y)+( c*exp(-2*beta*x^2)*z^2 + 2*beta*x*z + b^2*c)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added May 18, 2019.
Problem Chapter 6.3.2.18, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a e^{2 \alpha x^2} y^2 + 2 \alpha x y + a b^2 )w_y + \left ( c x^\beta z^2 + 2 \gamma x z + c d^2 x^\beta e^{2\gamma x^2} \right ) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*Exp[2*alpha*x^2]*y^2 + 2*alpha*x*y + a*b^2 )*D[w[x, y,z], y] +( c*x^beta*z^2 + 2*gamma*x*z + c*d^2*x^beta*Exp[2*gamma*x^2])*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)+ (a*exp(2*alpha*x^2)*y^2 + 2*alpha*x*y + a*b^2 )*diff(w(x,y,z),y)+( c*x^beta*z^2 + 2*gamma*x*z + c*d^2*x^beta*exp(2*gamma*x^2))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
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