Added June 27, 2019.
Problem Chapter 7.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + f(x,y) w_z = g(x,y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + f[x,y]*D[w[x,y,z],z]== g[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {g\left (K[2],y+\frac {b (K[2]-x)}{a}\right )}{a}dK[2]+c_1\left (y-\frac {b x}{a},z-\int _1^x\frac {f\left (K[1],y+\frac {b (K[1]-x)}{a}\right )}{a}dK[1]\right )\right \}\right \}\] Kernel message inconsistent or redundant transcendental equation
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ f(x,y)*diff(w(x,y,z),z)= g(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {g \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}, z -\left (\int _{}^{x}\frac {f \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )}{a}d\mathit {\_a} \right )\right )\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + f(x,y) g(z) w_z = h(x,y) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + f[x,y]*g[z]*D[w[x,y,z],z]== h[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ f(x,y)*g(z)*diff(w(x,y,z),z)= h(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {h \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}, \int \frac {a}{g \left (z \right )}d z -\left (\int _{}^{x}f \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )d\mathit {\_a} \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + (z+f(x,y)) w_z = g(x,y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + y*D[w[x, y,z], y] + (z+f[x,y])*D[w[x,y,z],z]== g[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {g\left (K[2],\frac {y K[2]}{x}\right )}{K[2]}dK[2]+c_1\left (\frac {y}{x},\frac {z}{x}-\int _1^x\frac {f\left (K[1],\frac {y K[1]}{x}\right )}{K[1]^2}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := x*diff(w(x,y,z),x)+ y*diff(w(x,y,z),y)+ (z+f(x,y))*diff(w(x,y,z),z)= g(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {g \left (\mathit {\_a} , \frac {\mathit {\_a} y}{x}\right )}{\mathit {\_a}}d\mathit {\_a} +\mathit {\_F1} \left (\frac {y}{x}, \frac {-x \left (\int _{}^{x}\frac {f \left (\mathit {\_a} , \frac {\mathit {\_a} y}{x}\right )}{\mathit {\_a}^{2}}d\mathit {\_a} \right )+z}{x}\right )\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + f(x,y) g(z) w_z = h(x,y) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] + f[x,y]*g[z]*D[w[x,y,z],z]== h[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+ f(x,y)*g(z)*diff(w(x,y,z),z)= h(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {h \left (\mathit {\_a} , y \mathit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )}{\mathit {\_a} a}d\mathit {\_a} +\mathit {\_F1} \left (y x^{-\frac {b}{a}}, \int \frac {a}{g \left (z \right )}d z -\left (\int _{}^{x}\frac {f \left (\mathit {\_a} , y \mathit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )}{\mathit {\_a}}d\mathit {\_a} \right )\right )\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)y+f_2(x)) w_y + (g_1(x,y) z+ g_2(x,y)) w_z = h(x,y,z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] + (g1[x,y]*z+g2[x,y])*D[w[x,y,z],z]== h[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 (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2],e^{-\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} z-\int _1^xe^{-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,x\}\right ]dK[3]} \text {g2}\left (K[4],e^{\int _1^{K[4]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[4]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )+\int _1^xh\left (K[5],e^{\int _1^{K[5]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[5]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right ),e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[5]}\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3],\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]-\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} \left (z-e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} \int _1^xe^{-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,x\}\right ]dK[3]} \text {g2}\left (K[4],e^{\int _1^{K[4]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[4]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[4]+e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} \int _1^{K[5]}e^{-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[5]}\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]} \text {g2}\left (K[4],e^{\int _1^{K[4]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[4]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )\right )dK[5]\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+ (g1(x,y)*z+g2(x,y))*diff(w(x,y,z),z)= h(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 ) = \int _{}^{x}h \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}, \left (z \,{\mathrm e}^{-\left (\int _{}^{x}\mathit {g1} \left (\mathit {\_f} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}+\int {\mathrm e}^{-\left (\int \mathit {g1} \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} -\left (\int _{}^{x}{\mathrm e}^{-\left (\int \mathit {g1} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d \mathit {\_a} \right )} \mathit {g2} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d\mathit {\_a} \right )\right ) {\mathrm e}^{\int \mathit {g1} \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h}}\right )d\mathit {\_h} +\mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z \,{\mathrm e}^{-\left (\int _{}^{x}\mathit {g1} \left (\mathit {\_f} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}-\left (\int _{}^{x}{\mathrm e}^{-\left (\int \mathit {g1} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d \mathit {\_a} \right )} \mathit {g2} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d\mathit {\_a} \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)y+f_2(x) y^k) w_y + (g_1(x,y) z+ g_2(x,y) z^m) w_z = h(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] + (g1[x,y]*z+g2[x,y]*z^m)*D[w[x,y,z],z]== h[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; local gamma; \ pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x,y)*z+g2(x,y)*z^m)*diff(w(x,y,z),z)= h(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 ) = \int _{}^{x}h \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}, \left (z^{-m +1} {\mathrm e}^{\left (m -1\right ) \left (\int _{}^{x}\mathit {g1} \left (\mathit {\_f} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}+\left (m -1\right ) \left (-\left (\int {\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )+\int _{}^{x}{\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d\mathit {\_h} \right )\right )^{-\frac {1}{m -1}} {\mathrm e}^{\int \mathit {g1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h}}\right )d\mathit {\_h} +\mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z^{-m +1} {\mathrm e}^{\left (m -1\right ) \left (\int _{}^{x}\mathit {g1} \left (\mathit {\_f} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}+\left (m -1\right ) \left (\int _{}^{x}{\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d\mathit {\_h} \right )\right )\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)y+f_2(x) y^k) w_y + (g_1(x,y)+ g_2(x,y) e^{\lambda z}) w_z = h(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] + (g1[x,y]+g2[x,y]*Exp[lambda*z])*D[w[x,y,z],z]== h[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)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x,y)+g2(x,y)*exp(lambda*z))*diff(w(x,y,z),z)= h(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 ) = \int _{}^{x}h \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}, \frac {\lambda \left (\int \mathit {g1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )+\ln \left (-\frac {1}{\left (\int {\mathrm e}^{\lambda \left (\int \mathit {g1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} -\left (\int _{}^{x}{\mathrm e}^{\lambda \left (\int \mathit {g1} \left (\mathit {\_b} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b} \right )} \mathit {f2} \left (\mathit {\_b} \right )d \mathit {\_b} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b}}\right )d \mathit {\_b} \right )} \mathit {g2} \left (\mathit {\_b} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b}}\right )d\mathit {\_b} \right )\right ) \lambda -{\mathrm e}^{\left (-z +\int _{}^{x}\mathit {g1} \left (\mathit {\_b} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b} \right )} \mathit {f2} \left (\mathit {\_b} \right )d \mathit {\_b} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b}}\right )d\mathit {\_b} \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_h} +\mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), \frac {-\lambda \left (\int _{}^{x}{\mathrm e}^{\lambda \left (\int \mathit {g1} \left (\mathit {\_b} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b} \right )} \mathit {f2} \left (\mathit {\_b} \right )d \mathit {\_b} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b}}\right )d \mathit {\_b} \right )} \mathit {g2} \left (\mathit {\_b} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b}}\right )d\mathit {\_b} \right )-{\mathrm e}^{\left (-z +\int _{}^{x}\mathit {g1} \left (\mathit {\_b} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b} \right )} \mathit {f2} \left (\mathit {\_b} \right )d \mathit {\_b} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_b} \right )d \mathit {\_b}}\right )d\mathit {\_b} \right ) \lambda }}{\lambda }\right )\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)+f_2(x) e^{\lambda y}) w_y + (g_1(x,y) z+ g_2(x,y) z^k) w_z = h(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] + (g1[x,y]*z+g2[x,y]*z^k)*D[w[x,y,z],z]== h[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)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g1(x,y)*z+g2(x,y)*z^k)*diff(w(x,y,z),z)= h(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 ) = \int _{}^{x}h \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }, {\mathrm e}^{\frac {\left (k -1\right ) \left (\int \mathit {g1} \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_h} \right )-\ln \left (z^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int _{}^{x}\mathit {g1} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_a} \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {g1} \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_h} \right )+\left (k -1\right ) \left (\int _{}^{x}{\mathrm e}^{\left (k -1\right ) \left (\int \mathit {g1} \left (\mathit {\_f} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_f} \right )} \mathit {g2} \left (\mathit {\_f} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_f} \right )\right )}{k -1}}\right )d\mathit {\_h} +\mathit {\_F1} \left (\frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )-{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}{\lambda }, z^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int _{}^{x}\mathit {g1} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_a} \right )}+\left (k -1\right ) \left (\int _{}^{x}{\mathrm e}^{\left (k -1\right ) \left (\int \mathit {g1} \left (\mathit {\_f} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_f} \right )} \mathit {g2} \left (\mathit {\_f} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_f} \right )\right )\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)+f_2(x) e^{\lambda y}) w_y + (g_1(x,y)+ g_2(x,y) e^{\beta z}) w_z = h(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] + (g1[x,y]+g2[x,y]*Exp[beta*z])*D[w[x,y,z],z]== h[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)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g1(x,y)+g2(x,y)*exp(beta*z))*diff(w(x,y,z),z)= h(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 ) = \int _{}^{x}h \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }, \frac {\beta \left (\int \mathit {g1} \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_h} \right )+\ln \left (\frac {1}{-\beta \left (\int {\mathrm e}^{\beta \left (\int \mathit {g1} \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_h} \right )+\beta \left (\int _{}^{x}{\mathrm e}^{\beta \left (\int \mathit {g1} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_a} \right )} \mathit {g2} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_a} \right )+{\mathrm e}^{\left (-z +\int _{}^{x}\mathit {g1} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_a} \right ) \beta }}\right )}{\beta }\right )d\mathit {\_h} +\mathit {\_F1} \left (\frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )-{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}{\lambda }, \frac {-\beta \left (\int _{}^{x}{\mathrm e}^{\beta \left (\int \mathit {g1} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d \mathit {\_a} \right )} \mathit {g2} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_a} \right )-{\mathrm e}^{\left (-z +\int _{}^{x}\mathit {g1} \left (\mathit {\_a} , \frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_a} \right ) \beta }}{\beta }\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.3.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + (h_1(x,y)+ h_2(x,y) z^m) w_z = h_3(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f1[x]*g1[y]*D[w[x, y,z], x] + f2[x]*g2[y]*D[w[x, y,z], y] + (h1[x,y]+h2[x,y]*z^m)*D[w[x,y,z],z]== h3[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; local gamma; pde := f1(x)*g1(y)*diff(w(x,y,z),x)+ f2(x)*g2(y)*diff(w(x,y,z),y)+ (h1(x,y)+h2(x,y)*z^m)*diff(w(x,y,z),z)= h3(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + (h_1(x,y)+ h_2(x,y) e^{\lambda z}) w_z = h_3(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f1[x]*g1[y]*D[w[x, y,z], x] + f2[x]*g2[y]*D[w[x, y,z], y] + (h1[x,y]+h2[x,y]*Exp[lambda*x])*D[w[x,y,z],z]== h3[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed