7.9.4 2.4

7.9.4.1 [1942] Problem 1

problem number 1942

Added Jan 16, 2020.

Problem Chapter 9.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c w_z = k x^n w + s x^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*D[w[x,y,z],z]==k*x^n*w[x,y,z]+ s*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {k x^{n+1}}{a n+a}} \left (-\frac {s x^{m+1} \left (\frac {k x^{n+1}}{a n+a}\right )^{-\frac {m+1}{n+1}} \operatorname {Gamma}\left (\frac {m+1}{n+1},\frac {k x^{n+1}}{a n+a}\right )}{a (n+1)}+c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {\left (\left (n +1\right )^{2} \left (k \,x^{n +1}+\left (m +n +2\right ) a \right ) s \,x^{m -n} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {-m -1}{n +1}} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {m +1}{n +1}} \left (\frac {k \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {-m -n -2}{2 n +2}} \WhittakerM \left (\frac {m -n}{2 n +2}, \frac {m +2 n +3}{2 n +2}, \frac {k \,x^{n +1}}{\left (n +1\right ) a}\right ) {\mathrm e}^{-\frac {k \,x^{n +1}}{2 \left (n +1\right ) a}}+\left (m +n +2\right ) \left (\left (n +1\right ) \left (m +n +2\right ) s \,x^{m -n} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {-m -1}{n +1}} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {m +1}{n +1}} \left (\frac {k \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {-m -n -2}{2 n +2}} \WhittakerM \left (\frac {m +n +2}{2 n +2}, \frac {m +2 n +3}{2 n +2}, \frac {k \,x^{n +1}}{\left (n +1\right ) a}\right ) {\mathrm e}^{-\frac {k \,x^{n +1}}{2 \left (n +1\right ) a}}+\left (m +1\right ) \left (m +2 n +3\right ) k \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a z -c x}{a}\right )\right ) a \right ) {\mathrm e}^{\frac {k \,x^{n +1}}{\left (n +1\right ) a}}}{\left (m +1\right ) \left (m +n +2\right ) \left (m +2 n +3\right ) a k}\]

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7.9.4.2 [1943] Problem 2

problem number 1943

Added Jan 16, 2020.

Problem Chapter 9.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a w_x + b y w_y + c z w_z = k x^n w + s x^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*y*D[w[x,y,z],y]+c*z*D[w[x,y,z],z]==k*x^n*w[x,y,z]+ s*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {k x^{n+1}}{a n+a}} \left (-\frac {s x^{m+1} \left (\frac {k x^{n+1}}{a n+a}\right )^{-\frac {m+1}{n+1}} \operatorname {Gamma}\left (\frac {m+1}{n+1},\frac {k x^{n+1}}{a n+a}\right )}{a (n+1)}+c_1\left (y e^{-\frac {b x}{a}},z e^{-\frac {c x}{a}}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {\left (\left (n +1\right )^{2} \left (k \,x^{n +1}+\left (m +n +2\right ) a \right ) s \,x^{m -n} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {-m -1}{n +1}} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {m +1}{n +1}} \left (\frac {k \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {-m -n -2}{2 n +2}} \WhittakerM \left (\frac {m -n}{2 n +2}, \frac {m +2 n +3}{2 n +2}, \frac {k \,x^{n +1}}{\left (n +1\right ) a}\right ) {\mathrm e}^{-\frac {k \,x^{n +1}}{2 \left (n +1\right ) a}}+\left (m +n +2\right ) \left (\left (n +1\right ) \left (m +n +2\right ) s \,x^{m -n} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {-m -1}{n +1}} \left (\frac {k}{\left (n +1\right ) a}\right )^{\frac {m +1}{n +1}} \left (\frac {k \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {-m -n -2}{2 n +2}} \WhittakerM \left (\frac {m +n +2}{2 n +2}, \frac {m +2 n +3}{2 n +2}, \frac {k \,x^{n +1}}{\left (n +1\right ) a}\right ) {\mathrm e}^{-\frac {k \,x^{n +1}}{2 \left (n +1\right ) a}}+\left (m +1\right ) \left (m +2 n +3\right ) k \textit {\_F1} \left (y \,{\mathrm e}^{-\frac {b x}{a}}, z \,{\mathrm e}^{-\frac {c x}{a}}\right )\right ) a \right ) {\mathrm e}^{\frac {k \,x^{n +1}}{\left (n +1\right ) a}}}{\left (m +1\right ) \left (m +n +2\right ) \left (m +2 n +3\right ) a k}\]

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7.9.4.3 [1944] Problem 3

problem number 1944

Added Jan 16, 2020.

Problem Chapter 9.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a z w_y + b y w_z = c x^n w + s x^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+a*z*D[w[x,y,z],y]+b*y*D[w[x,y,z],z]==c*x^n*w[x,y,z]+ s*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{n+1}}{n+1}} \left (-\frac {s x^{m+1} \left (\frac {c x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \operatorname {Gamma}\left (\frac {m+1}{n+1},\frac {c x^{n+1}}{n+1}\right )}{n+1}+c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \left (\int _{}^{y}\frac {s \left (\frac {\sqrt {a b}\, x +\ln \left (\frac {\textit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )^{m} {\mathrm e}^{-c \left (\int \frac {\left (\frac {\sqrt {a b}\, x +\ln \left (\frac {\textit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )^{n}}{\sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_b} \right )}}{\sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_b} +\textit {\_F1} \left (\frac {a \,z^{2}-b \,y^{2}}{a}, -\frac {-\sqrt {a b}\, x +\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\right ) {\mathrm e}^{\int _{}^{y}\frac {c \left (\frac {\sqrt {a b}\, x +\ln \left (\frac {\textit {\_a} a b +\sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )^{n}}{\sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_a}}\]

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7.9.4.4 [1945] Problem 4

problem number 1945

Added Jan 16, 2020.

Problem Chapter 9.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a x^n w_y + b x^m w_z = c x^k w + s x^r \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+a*x^n*D[w[x,y,z],y]+b*x^m*D[w[x,y,z],z]==c*x^k*w[x,y,z]+ s*x^r; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{k+1}}{k+1}} \left (-\frac {s x^{r+1} \left (\frac {c x^{k+1}}{k+1}\right )^{-\frac {r+1}{k+1}} \operatorname {Gamma}\left (\frac {r+1}{k+1},\frac {c x^{k+1}}{k+1}\right )}{k+1}+c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ a*x^n*diff(w(x,y,z),y)+ b*x^m*diff(w(x,y,z),z)=c*x^k*w(x,y,z)+ s*x^r; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {\left (\left (k +1\right ) \left (\left (k +1\right ) \left (c \,x^{k +1}+k +r +2\right ) \WhittakerM \left (\frac {-k +r}{2 k +2}, \frac {2 k +r +3}{2 k +2}, \frac {c \,x^{k +1}}{k +1}\right )+\left (k +r +2\right )^{2} \WhittakerM \left (\frac {k +r +2}{2 k +2}, \frac {2 k +r +3}{2 k +2}, \frac {c \,x^{k +1}}{k +1}\right )\right ) s \,x^{-k +r} \left (\frac {c}{k +1}\right )^{\frac {-r -1}{k +1}} \left (\frac {c}{k +1}\right )^{\frac {r +1}{k +1}} \left (\frac {c \,x^{k +1}}{k +1}\right )^{\frac {-k -r -2}{2 k +2}} {\mathrm e}^{-\frac {c \,x^{k +1}}{2 k +2}}+2 \left (k +r +2\right ) \left (r +1\right ) \left (k +\frac {r}{2}+\frac {3}{2}\right ) c \textit {\_F1} \left (\frac {-a x \,x^{n}+\left (n +1\right ) y}{n +1}, \frac {-b x \,x^{m}+\left (m +1\right ) z}{m +1}\right )\right ) {\mathrm e}^{\frac {c \,x^{k +1}}{k +1}}}{\left (r +1\right ) \left (k +r +2\right ) \left (2 k +r +3\right ) c}\]

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7.9.4.5 [1946] Problem 5

problem number 1946

Added Jan 16, 2020.

Problem Chapter 9.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + b x^n w_y + c y^m w_z = a w + s x^k \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+b*x^n*D[w[x,y,z],y]+c*x^m*D[w[x,y,z],z]==a*w[x,y,z]+ s*x^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{a x} \left (-\frac {s x^k (a x)^{-k} \operatorname {Gamma}(k+1,a x)}{a}+c_1\left (\frac {-b x^{n+1}+n y+y}{n+1},\frac {-c x^{m+1}+m z+z}{m+1}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {\left (s \,x^{k} \left (a x \right )^{-\frac {k}{2}} \WhittakerM \left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, a x \right ) {\mathrm e}^{-\frac {a x}{2}}+\left (k +1\right ) a \textit {\_F1} \left (\frac {-b x \,x^{n}+\left (n +1\right ) y}{n +1}, \frac {-c x \,x^{m}+\left (m +1\right ) z}{m +1}\right )\right ) {\mathrm e}^{a x}}{\left (k +1\right ) a}\]

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7.9.4.6 [1947] Problem 6

problem number 1947

Added Jan 16, 2020.

Problem Chapter 9.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a y + \beta x^n) w_y + (b z + \gamma x^m) w_z = c x^k w + s x^r \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{k+1}}{k+1}} \left (-\frac {s x^{r+1} \left (\frac {c x^{k+1}}{k+1}\right )^{-\frac {r+1}{k+1}} \operatorname {Gamma}\left (\frac {r+1}{k+1},\frac {c x^{k+1}}{k+1}\right )}{k+1}+c_1\left (\gamma b^{-m-1} \operatorname {Gamma}(m+1,b x)+z e^{-b x},\beta a^{-n-1} \operatorname {Gamma}(n+1,a x)+y e^{-a x}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {\left (\left (k +1\right ) \left (\left (k +1\right ) \left (c \,x^{k +1}+k +r +2\right ) \WhittakerM \left (\frac {-k +r}{2 k +2}, \frac {2 k +r +3}{2 k +2}, \frac {c \,x^{k +1}}{k +1}\right )+\left (k +r +2\right )^{2} \WhittakerM \left (\frac {k +r +2}{2 k +2}, \frac {2 k +r +3}{2 k +2}, \frac {c \,x^{k +1}}{k +1}\right )\right ) s \,x^{-k +r} \left (\frac {c}{k +1}\right )^{\frac {-r -1}{k +1}} \left (\frac {c}{k +1}\right )^{\frac {r +1}{k +1}} \left (\frac {c \,x^{k +1}}{k +1}\right )^{\frac {-k -r -2}{2 k +2}} {\mathrm e}^{-\frac {c \,x^{k +1}}{2 k +2}}+2 \left (k +r +2\right ) \left (r +1\right ) \left (k +\frac {r}{2}+\frac {3}{2}\right ) c \textit {\_F1} \left (\frac {\left (-\beta \,x^{n} \left (a x \right )^{-\frac {n}{2}} \WhittakerM \left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, a x \right ) {\mathrm e}^{\frac {a x}{2}}+\left (n +1\right ) a y \right ) {\mathrm e}^{-a x}}{\left (n +1\right ) a}, \frac {\left (-\gamma \,x^{m} \left (b x \right )^{-\frac {m}{2}} \WhittakerM \left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, b x \right ) {\mathrm e}^{\frac {b x}{2}}+\left (m +1\right ) b z \right ) {\mathrm e}^{-b x}}{\left (m +1\right ) b}\right )\right ) {\mathrm e}^{\frac {c \,x^{k +1}}{k +1}}}{\left (r +1\right ) \left (k +r +2\right ) \left (2 k +r +3\right ) c}\]

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7.9.4.7 [1948] Problem 7

problem number 1948

Added Jan 16, 2020.

Problem Chapter 9.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^{n_1} y + a_2 x^{n_2}) w_y + (b_1 y^{m_1} z + b_2 y^{m_2}) w_z = c w + s_1 x y^{k_1}+ s_2 x^{k_2} z \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x^n1*y + a2*x^n2)*D[w[x,y,z],y]+(b1*y^m1*z + b2*y^m2)*D[w[x,y,z],z]==c*w[x,y,z]+ s1*x*y^k1+ s2*x^k2*z; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{c x} \left (c_1\left (\text {a2} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}}+e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} y,e^{-\int _1^x\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]} z-\int _1^x\text {b2} e^{-\int _1^{K[2]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^x\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]}\right ),\{K[1],1,x\}\right ]dK[1]} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[2]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m2}}dK[2]\right )+\int _1^xe^{-c K[3]} \left (\text {s1} K[3] \left (\frac {\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[3]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} \left ((\text {n1}+1) y \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}}+\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[3]^{\text {n1}+1}}{\text {n1}+1}\right )\right )}{\text {n1}+1}\right )^{\text {k1}}+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[2]}\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[2]}\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[2]}\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[2]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[3]}\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1],\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]-\int _1^x\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]} \text {s2} z K[3]^{\text {k2}}-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[2]}\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[2]}\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[2]}\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[2]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[3]}\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1],\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]} \text {s2} K[3]^{\text {k2}} \int _1^x\text {b2} e^{-\int _1^{K[2]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^x\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]}\right ),\{K[1],1,x\}\right ]dK[1]} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[2]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m2}}dK[2]+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[2]}\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[2]}\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[2]}\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[2]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[3]}\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1],\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]} \text {s2} K[3]^{\text {k2}} \int _1^{K[3]}\text {b2} e^{-\int _1^{K[2]}\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[2]}\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[2]}\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[2]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[3]}\text {b1} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]}\right ),\{K[1],1,K[2]\}\right ],\{K[1],1,K[3]\}\right ]}\right ),\{K[1],1,K[3]\}\right ]dK[1]} \left (\text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[2]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {n1}+\text {n2}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {a2} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m2}}dK[2]\right )dK[3]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*x^(n__1)*y + a__2*x^(n__2))*diff(w(x,y,z),y)+ (b__1*y^(m__1)*z + b__2*y^(m__2))*diff(w(x,y,z),z)=c*w(x,y,z)+ s__1*x*y^(k__1)+ s__2*x^(k__2)*z; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[\text {Expression too large to display}\]

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7.9.4.8 [1949] Problem 8

problem number 1949

Added Jan 16, 2020.

Problem Chapter 9.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^{\lambda _1} y + a_2 x^{\lambda _2} y^k) w_y + (b_1 x^{\beta _1} z + b_2 x^{\beta _2} z^m) w_z = c_1 x^{\gamma _1} w + c_2 y^{\gamma _2} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x^lambda1*y + a2*x^lambda2*y^k)*D[w[x,y,z],y]+(b1*x^beta1*z + b2*x^beta2*z^m)*D[w[x,y,z],z]==c1*x^gamma1*w[x,y,z]+ c2*y^gamma2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {c1} x^{\text {gamma1}+1}}{\text {gamma1}+1}} \left (\int _1^x\text {c2} e^{-\frac {\text {c1} K[1]^{\text {gamma1}+1}}{\text {gamma1}+1}} \left (\left (\frac {(-1)^{-\frac {\text {lambda2}+1}{\text {lambda1}+1}} \text {a1}^{-\frac {\text {lambda2}+1}{\text {lambda1}+1}} \exp \left (-\frac {\text {a1} \left (x^{\text {lambda1}+1}+(k-1) K[1]^{\text {lambda1}+1}\right )}{\text {lambda1}+1}\right ) (k-1)^{-\frac {\text {lambda2}+1}{\text {lambda1}+1}} y^{-k} \left (-\text {a2} e^{\frac {\text {a1} x^{\text {lambda1}+1}}{\text {lambda1}+1}} (k-1) (\text {lambda1}+1)^{\frac {\text {lambda2}+1}{\text {lambda1}+1}} \operatorname {Gamma}\left (\frac {\text {lambda2}+1}{\text {lambda1}+1},-\frac {\text {a1} (k-1) x^{\text {lambda1}+1}}{\text {lambda1}+1}\right ) y^k+\text {a2} e^{\frac {\text {a1} x^{\text {lambda1}+1}}{\text {lambda1}+1}} (k-1) (\text {lambda1}+1)^{\frac {\text {lambda2}+1}{\text {lambda1}+1}} \operatorname {Gamma}\left (\frac {\text {lambda2}+1}{\text {lambda1}+1},-\frac {\text {a1} (k-1) K[1]^{\text {lambda1}+1}}{\text {lambda1}+1}\right ) y^k+(-1)^{\frac {\text {lambda2}+1}{\text {lambda1}+1}} \text {a1}^{\frac {\text {lambda2}+1}{\text {lambda1}+1}} e^{\frac {\text {a1} k x^{\text {lambda1}+1}}{\text {lambda1}+1}} (k-1)^{\frac {\text {lambda2}+1}{\text {lambda1}+1}} (\text {lambda1}+1) y\right )}{\text {lambda1}+1}\right )^{\frac {1}{1-k}}\right )^{\text {gamma2}}dK[1]+c_1\left (\text {b2} (-1)^{\frac {\text {beta1}-\text {beta2}}{\text {beta1}+1}} (\text {beta1}+1)^{\frac {\text {beta2}-\text {beta1}}{\text {beta1}+1}} \text {b1}^{-\frac {\text {beta2}+1}{\text {beta1}+1}} (m-1)^{\frac {\text {beta1}-\text {beta2}}{\text {beta1}+1}} \operatorname {Gamma}\left (\frac {\text {beta2}+1}{\text {beta1}+1},-\frac {\text {b1} (m-1) x^{\text {beta1}+1}}{\text {beta1}+1}\right )+z^{1-m} e^{\frac {\text {b1} (m-1) x^{\text {beta1}+1}}{\text {beta1}+1}},\text {a2} (-1)^{\frac {\text {lambda1}-\text {lambda2}}{\text {lambda1}+1}} (\text {lambda1}+1)^{\frac {\text {lambda2}-\text {lambda1}}{\text {lambda1}+1}} \text {a1}^{-\frac {\text {lambda2}+1}{\text {lambda1}+1}} (k-1)^{\frac {\text {lambda1}-\text {lambda2}}{\text {lambda1}+1}} \operatorname {Gamma}\left (\frac {\text {lambda2}+1}{\text {lambda1}+1},-\frac {\text {a1} (k-1) x^{\text {lambda1}+1}}{\text {lambda1}+1}\right )+y^{1-k} e^{\frac {\text {a1} (k-1) x^{\text {lambda1}+1}}{\text {lambda1}+1}}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*x^(lambda__1)*y + a__2*x^(lambda__2)*y^k)*diff(w(x,y,z),y)+ (b__1*x^(beta__1)*z + b__2*x^(beta__2)*z^m)*diff(w(x,y,z),z)=c__1*x^(gamma__1)*w(x,y,z)+ c__2*y^(gamma__2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \left (\int _{}^{x}c_{2} \left (\left (\frac {-\left (\lambda _{1} +1\right )^{2} \left (\left (k -1\right ) a_{1} \textit {\_a}^{\lambda _{2} +1}+\left (-\lambda _{1} -\lambda _{2} -2\right ) \textit {\_a}^{-\lambda _{1} +\lambda _{2}}\right ) a_{2} \left (-\frac {\left (k -1\right ) a_{1} \textit {\_a}^{\lambda _{1} +1}}{\lambda _{1} +1}\right )^{\frac {-\lambda _{1} -\lambda _{2} -2}{2 \lambda _{1} +2}} \WhittakerM \left (\frac {-\lambda _{1} +\lambda _{2}}{2 \lambda _{1} +2}, \frac {2 \lambda _{1} +\lambda _{2} +3}{2 \lambda _{1} +2}, -\frac {\left (k -1\right ) a_{1} \textit {\_a}^{\lambda _{1} +1}}{\lambda _{1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) a_{1} \textit {\_a}^{\lambda _{1} +1}}{2 \lambda _{1} +2}}+\left (\lambda _{1} +1\right )^{2} \left (\left (k -1\right ) a_{1} x^{\lambda _{2} +1}+\left (-\lambda _{1} -\lambda _{2} -2\right ) x^{-\lambda _{1} +\lambda _{2}}\right ) a_{2} \left (-\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right )^{\frac {-\lambda _{1} -\lambda _{2} -2}{2 \lambda _{1} +2}} \WhittakerM \left (\frac {-\lambda _{1} +\lambda _{2}}{2 \lambda _{1} +2}, \frac {2 \lambda _{1} +\lambda _{2} +3}{2 \lambda _{1} +2}, -\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{2 \lambda _{1} +2}}-\left (-\left (\lambda _{1} +1\right ) \left (\lambda _{1} +\lambda _{2} +2\right ) a_{2} \textit {\_a}^{-\lambda _{1} +\lambda _{2}} \left (-\frac {\left (k -1\right ) a_{1} \textit {\_a}^{\lambda _{1} +1}}{\lambda _{1} +1}\right )^{\frac {-\lambda _{1} -\lambda _{2} -2}{2 \lambda _{1} +2}} \WhittakerM \left (\frac {\lambda _{1} +\lambda _{2} +2}{2 \lambda _{1} +2}, \frac {2 \lambda _{1} +\lambda _{2} +3}{2 \lambda _{1} +2}, -\frac {\left (k -1\right ) a_{1} \textit {\_a}^{\lambda _{1} +1}}{\lambda _{1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) a_{1} \textit {\_a}^{\lambda _{1} +1}}{2 \lambda _{1} +2}}+\left (\lambda _{1} +1\right ) \left (\lambda _{1} +\lambda _{2} +2\right ) a_{2} x^{-\lambda _{1} +\lambda _{2}} \left (-\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right )^{\frac {-\lambda _{1} -\lambda _{2} -2}{2 \lambda _{1} +2}} \WhittakerM \left (\frac {\lambda _{1} +\lambda _{2} +2}{2 \lambda _{1} +2}, \frac {2 \lambda _{1} +\lambda _{2} +3}{2 \lambda _{1} +2}, -\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{2 \lambda _{1} +2}}-2 \left (\lambda _{2} +1\right ) \left (\lambda _{1} +\frac {\lambda _{2}}{2}+\frac {3}{2}\right ) a_{1} y^{-k +1} {\mathrm e}^{\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}}\right ) \left (\lambda _{1} +\lambda _{2} +2\right )}{\left (2 \lambda _{1} +\lambda _{2} +3\right ) \left (\lambda _{1} +\lambda _{2} +2\right ) \left (\lambda _{2} +1\right ) a_{1}}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\frac {a_{1} \textit {\_a}^{\lambda _{1} +1}}{\lambda _{1} +1}}\right )^{\gamma _{2}} {\mathrm e}^{-\frac {c_{1} \textit {\_a}^{\gamma _{1} +1}}{\gamma _{1} +1}}d \textit {\_a} +\textit {\_F1} \left (\frac {\left (\lambda _{1} +1\right )^{2} \left (\left (k -1\right ) a_{1} x^{\lambda _{2} +1}+\left (-\lambda _{1} -\lambda _{2} -2\right ) x^{-\lambda _{1} +\lambda _{2}}\right ) a_{2} \left (-\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right )^{\frac {-\lambda _{1} -\lambda _{2} -2}{2 \lambda _{1} +2}} \WhittakerM \left (\frac {-\lambda _{1} +\lambda _{2}}{2 \lambda _{1} +2}, \frac {2 \lambda _{1} +\lambda _{2} +3}{2 \lambda _{1} +2}, -\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{2 \lambda _{1} +2}}-\left (\lambda _{1} +\lambda _{2} +2\right ) \left (\left (\lambda _{1} +1\right ) \left (\lambda _{1} +\lambda _{2} +2\right ) a_{2} x^{-\lambda _{1} +\lambda _{2}} \left (-\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right )^{\frac {-\lambda _{1} -\lambda _{2} -2}{2 \lambda _{1} +2}} \WhittakerM \left (\frac {\lambda _{1} +\lambda _{2} +2}{2 \lambda _{1} +2}, \frac {2 \lambda _{1} +\lambda _{2} +3}{2 \lambda _{1} +2}, -\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{2 \lambda _{1} +2}}-2 \left (\lambda _{2} +1\right ) \left (\lambda _{1} +\frac {\lambda _{2}}{2}+\frac {3}{2}\right ) a_{1} y^{-k +1} {\mathrm e}^{\frac {\left (k -1\right ) a_{1} x^{\lambda _{1} +1}}{\lambda _{1} +1}}\right )}{\left (2 \lambda _{1} +\lambda _{2} +3\right ) \left (\lambda _{2} +1\right ) \left (\lambda _{1} +\lambda _{2} +2\right ) a_{1}}, \frac {\left (\beta _{1} +1\right )^{2} \left (\left (m -1\right ) b_{1} x^{\beta _{2} +1}+\left (-\beta _{1} -\beta _{2} -2\right ) x^{-\beta _{1} +\beta _{2}}\right ) b_{2} \left (-\frac {\left (m -1\right ) b_{1} x^{\beta _{1} +1}}{\beta _{1} +1}\right )^{\frac {-\beta _{1} -\beta _{2} -2}{2 \beta _{1} +2}} \WhittakerM \left (\frac {-\beta _{1} +\beta _{2}}{2 \beta _{1} +2}, \frac {2 \beta _{1} +\beta _{2} +3}{2 \beta _{1} +2}, -\frac {\left (m -1\right ) b_{1} x^{\beta _{1} +1}}{\beta _{1} +1}\right ) {\mathrm e}^{\frac {\left (m -1\right ) b_{1} x^{\beta _{1} +1}}{2 \beta _{1} +2}}-\left (\beta _{1} +\beta _{2} +2\right ) \left (\left (\beta _{1} +1\right ) \left (\beta _{1} +\beta _{2} +2\right ) b_{2} x^{-\beta _{1} +\beta _{2}} \left (-\frac {\left (m -1\right ) b_{1} x^{\beta _{1} +1}}{\beta _{1} +1}\right )^{\frac {-\beta _{1} -\beta _{2} -2}{2 \beta _{1} +2}} \WhittakerM \left (\frac {\beta _{1} +\beta _{2} +2}{2 \beta _{1} +2}, \frac {2 \beta _{1} +\beta _{2} +3}{2 \beta _{1} +2}, -\frac {\left (m -1\right ) b_{1} x^{\beta _{1} +1}}{\beta _{1} +1}\right ) {\mathrm e}^{\frac {\left (m -1\right ) b_{1} x^{\beta _{1} +1}}{2 \beta _{1} +2}}-2 \left (\beta _{2} +1\right ) \left (\beta _{1} +\frac {\beta _{2}}{2}+\frac {3}{2}\right ) b_{1} z^{-m +1} {\mathrm e}^{\frac {\left (m -1\right ) b_{1} x^{\beta _{1} +1}}{\beta _{1} +1}}\right )}{\left (2 \beta _{1} +\beta _{2} +3\right ) \left (\beta _{2} +1\right ) \left (\beta _{1} +\beta _{2} +2\right ) b_{1}}\right )\right ) {\mathrm e}^{\frac {c_{1} x^{\gamma _{1} +1}}{\gamma _{1} +1}}\]

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7.9.4.9 [1950] Problem 9

problem number 1950

Added Jan 16, 2020.

Problem Chapter 9.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^{\lambda _1} y + a_2 x^{\lambda _2} y^k) w_y + (b_1 y^{\beta _1} z + b_2 y^{\beta _2} z^m) w_z = c_1 x^{\gamma _1} w + c_2 z^{\gamma _2} \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x^lambda1*y + a2*x^lambda2*y^k)*D[w[x,y,z],y]+(b1*y^beta1*z + b2*y^beta2*z^m)*D[w[x,y,z],z]==c1*x^gamma1*w[x,y,z]+ c2*z^gamma2; 
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)+ (a__1*x^(lambda__1)*y + a__2*x^(lambda__2)*y^k)*diff(w(x,y,z),y)+ (b__1*y^(beta__1)*z + b__2*y^(beta__2)*z^m)*diff(w(x,y,z),z)=c__1*x^(gamma__1)*w(x,y,z)+ c__2*z^(gamma__2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[\text {Expression too large to display}\]

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7.9.4.10 [1951] Problem 10

problem number 1951

Added Jan 16, 2020.

Problem Chapter 9.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ x w_x + a y w_y + b z w_z = c x^n w + k x^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x,y,z],x]+a*y*D[w[x,y,z],y]+b*z*D[w[x,y,z],z]==c*x^n*w[x,y,z]+ k*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^n}{n}} \left (-\frac {k x^m \left (\frac {c x^n}{n}\right )^{-\frac {m}{n}} \operatorname {Gamma}\left (\frac {m}{n},\frac {c x^n}{n}\right )}{n}+c_1\left (y x^{-a},z x^{-b}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {\left (\left (c \,x^{n}+m +n \right ) k \,n^{2} x^{m -n} \left (\frac {c}{n}\right )^{\frac {m}{n}} \left (\frac {c}{n}\right )^{-\frac {m}{n}} \left (\frac {c \,x^{n}}{n}\right )^{-\frac {m +n}{2 n}} \WhittakerM \left (\frac {m -n}{2 n}, \frac {m +2 n}{2 n}, \frac {c \,x^{n}}{n}\right ) {\mathrm e}^{-\frac {c \,x^{n}}{2 n}}+\left (\left (m +n \right ) k n \,x^{m -n} \left (\frac {c}{n}\right )^{\frac {m}{n}} \left (\frac {c}{n}\right )^{-\frac {m}{n}} \left (\frac {c \,x^{n}}{n}\right )^{-\frac {m +n}{2 n}} \WhittakerM \left (\frac {m +n}{2 n}, \frac {m +2 n}{2 n}, \frac {c \,x^{n}}{n}\right ) {\mathrm e}^{-\frac {c \,x^{n}}{2 n}}+\left (m +2 n \right ) c m \textit {\_F1} \left (y \,x^{-a}, z \,x^{-b}\right )\right ) \left (m +n \right )\right ) {\mathrm e}^{\frac {c \,x^{n}}{n}}}{\left (m +n \right ) \left (m +2 n \right ) c m}\]

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7.9.4.11 [1952] Problem 11

problem number 1952

Added Jan 16, 2020.

Problem Chapter 9.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ x w_x + a z w_y + b y w_z = c x^n w + k x^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x,y,z],x]+a*z*D[w[x,y,z],y]+b*y*D[w[x,y,z],z]==c*x^n*w[x,y,z]+ k*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^n}{n}} \left (-\frac {k x^m \left (\frac {c x^n}{n}\right )^{-\frac {m}{n}} \operatorname {Gamma}\left (\frac {m}{n},\frac {c x^n}{n}\right )}{n}+c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \left (\int _{}^{y}\frac {k \left (x \left (\frac {\textit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )^{\frac {1}{\sqrt {a b}}} \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right )^{m} {\mathrm e}^{-c \left (\int \frac {\left (x \left (\frac {\textit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )^{\frac {1}{\sqrt {a b}}} \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right )^{n}}{\sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_b} \right )}}{\sqrt {\left (a \,z^{2}+\left (\textit {\_b}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_b} +\textit {\_F1} \left (\frac {a \,z^{2}-b \,y^{2}}{a}, x \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right )\right ) {\mathrm e}^{\int _{}^{y}\frac {c \left (x \left (\frac {\textit {\_a} a b +\sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )^{\frac {1}{\sqrt {a b}}} \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right )^{n}}{\sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_a}}\]

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7.9.4.12 [1953] Problem 12

problem number 1953

Added Jan 16, 2020.

Problem Chapter 9.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ b c x w_x + c(b y + c z) w_y + b(b y - c z) w_z = k x^n w + s x^m \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  b*c*x*D[w[x,y,z],x]+ c*(b*y + c*z)*D[w[x,y,z],y]+b*(b*y - c*z)*D[w[x,y,z],z]==k*x^n*w[x,y,z]+ s*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  b*c*x*diff(w(x,y,z),x)+ c*(b*y + c*z)*diff(w(x,y,z),y)+ b*(b*y - c*z)*diff(w(x,y,z),z)=k*x^n*w(x,y,z)+ s*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \left (-\left (\int _{}^{y}-\frac {s \left (x \left (\frac {\frac {\sqrt {2}\, \textit {\_b} \,b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+\sqrt {\frac {2 \textit {\_b}^{2} b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+1}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}{\sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}\right )^{\frac {\sqrt {2}\, b}{2 \sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}} \left (\frac {\frac {\sqrt {2}\, b^{2} y}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+\left (\frac {b y}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}+\frac {c z}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}\right ) \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}{\sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, b}{2 \sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}}\right )^{m} {\mathrm e}^{-\frac {k \left (\int \frac {\left (x \left (\frac {\frac {\sqrt {2}\, \textit {\_b} \,b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+\sqrt {\frac {2 \textit {\_b}^{2} b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+1}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}{\sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}\right )^{\frac {\sqrt {2}\, b}{2 \sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}} \left (\frac {\frac {\sqrt {2}\, b^{2} y}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+\left (\frac {b y}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}+\frac {c z}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}\right ) \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}{\sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, b}{2 \sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}}\right )^{n}}{\sqrt {\frac {2 \textit {\_b}^{2} b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+1}}d \textit {\_b} \right )}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, c}}}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {2 \textit {\_b}^{2} b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+1}\, c}d \textit {\_b} \right )+\textit {\_F1} \left (-\frac {1}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}, x \left (\frac {\frac {\sqrt {2}\, b^{2} y}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+\left (\frac {b y}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}+\frac {c z}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}\right ) \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}{\sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, b}{2 \sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}}\right )\right ) {\mathrm e}^{-\left (\int _{}^{y}-\frac {k \left (x \left (\frac {\frac {\sqrt {2}\, \textit {\_a} \,b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+\sqrt {\frac {2 \textit {\_a}^{2} b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+1}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}{\sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}\right )^{\frac {\sqrt {2}\, b}{2 \sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}} \left (\frac {\frac {\sqrt {2}\, b^{2} y}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+\left (\frac {b y}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}+\frac {c z}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}\right ) \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}{\sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, b}{2 \sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}}}}\right )^{n}}{\sqrt {-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}\, \sqrt {\frac {2 \textit {\_a}^{2} b^{2}}{-b^{2} y^{2}+2 b c z y +c^{2} z^{2}}+1}\, c}d \textit {\_a} \right )}\]

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7.9.4.13 [1954] Problem 13

problem number 1954

Added Jan 16, 2020.

Problem Chapter 9.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ b_1 x^{n_1} w_x + b_2 y^{n_2} w_y + b_3 z^{n_3} w_z = a w + c_1 x^{k_1} + c_2 y^{k_2} + c_3 x^{k_3} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  b1*x^n1*D[w[x,y,z],x]+ b2*y^n2*D[w[x,y,z],y]+b3*z^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*x^k1+c2*y^k2+c3*x^k3; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {a x^{1-\text {n1}}}{\text {b1}-\text {b1} \text {n1}}} \left (\int _1^x\frac {e^{\frac {a K[1]^{1-\text {n1}}}{\text {b1} (\text {n1}-1)}} K[1]^{-\text {n1}} \left (\text {c1} K[1]^{\text {k1}}+\text {c3} K[1]^{\text {k3}}+\text {c2} \left (\left (\frac {\text {b2} (\text {n2}-1) x^{-\text {n1}} \left (x^{\text {n1}} K[1]-x K[1]^{\text {n1}}\right ) K[1]^{-\text {n1}}}{\text {b1} (\text {n1}-1)}+\left (\frac {1}{y}\right )^{\text {n2}-1}\right )^{\frac {1}{1-\text {n2}}}\right )^{\text {k2}}\right )}{\text {b1}}dK[1]+c_1\left (\frac {\text {b2} x^{1-\text {n1}}}{\text {b1} (\text {n1}-1)}-\frac {\left (\frac {1}{y}\right )^{\text {n2}-1}}{\text {n2}-1},\frac {\text {b3} x^{1-\text {n1}}}{\text {b1} (\text {n1}-1)}-\frac {\left (\frac {1}{z}\right )^{\text {n3}-1}}{\text {n3}-1}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=   b__1*x^(n__1)*diff(w(x,y,z),x)+ b__2*y^(n__2)*diff(w(x,y,z),y)+ b__3*z^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*x^(k__1)+c__2*y^(k__2)+c__3*x^(k__3); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \left (\int _{}^{x}\frac {\left (c_{2} \textit {\_a}^{-n_{1}} \left (\left (\frac {\left (n_{1} -1\right ) b_{1} y^{-n_{2} +1}+\left (n_{2} -1\right ) b_{2} \textit {\_a}^{-n_{1} +1}-\left (n_{2} -1\right ) b_{2} x^{-n_{1} +1}}{\left (n_{1} -1\right ) b_{1}}\right )^{-\frac {1}{n_{2} -1}}\right )^{k_{2}}+c_{1} \textit {\_a}^{k_{1} -n_{1}}+c_{3} \textit {\_a}^{k_{3} -n_{1}}\right ) {\mathrm e}^{\frac {a \,\textit {\_a}^{-n_{1} +1}}{\left (n_{1} -1\right ) b_{1}}}}{b_{1}}d \textit {\_a} +\textit {\_F1} \left (\frac {\left (n_{1} -1\right ) b_{1} y^{-n_{2} +1}-\left (n_{2} -1\right ) b_{2} x^{-n_{1} +1}}{\left (n_{1} -1\right ) b_{1}}, \frac {\left (n_{1} -1\right ) b_{1} z^{-n_{3} +1}-\left (n_{3} -1\right ) b_{3} x^{-n_{1} +1}}{\left (n_{1} -1\right ) b_{1}}\right )\right ) {\mathrm e}^{-\frac {a \,x^{-n_{1} +1}}{\left (n_{1} -1\right ) b_{1}}}\]

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7.9.4.14 [1955] Problem 14

problem number 1955

Added Jan 16, 2020.

Problem Chapter 9.2.4.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a_1 x^{n_1} w_x + a_2 y^{n_2} w_y + a_3 z^{n_3} w_z = b x^k w +c x^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a1*x^n1*D[w[x,y,z],x]+ a2*y^n2*D[w[x,y,z],y]+a3*z^n3*D[w[x,y,z],z]==b*x^k*w[x,y,z]+ c*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {b x^{k-\text {n1}+1}}{\text {a1} k-\text {a1} \text {n1}+\text {a1}}} \left (-\frac {c x^{m-\text {n1}+1} \left (\frac {b x^{k-\text {n1}+1}}{\text {a1} k-\text {a1} \text {n1}+\text {a1}}\right )^{\frac {-m+\text {n1}-1}{k-\text {n1}+1}} \operatorname {Gamma}\left (\frac {m-\text {n1}+1}{k-\text {n1}+1},\frac {b x^{k-\text {n1}+1}}{\text {a1} k-\text {a1} \text {n1}+\text {a1}}\right )}{\text {a1} (k-\text {n1}+1)}+c_1\left (\frac {\text {a2} x^{1-\text {n1}}}{\text {a1} (\text {n1}-1)}-\frac {\left (\frac {1}{y}\right )^{\text {n2}-1}}{\text {n2}-1},\frac {\text {a3} x^{1-\text {n1}}}{\text {a1} (\text {n1}-1)}-\frac {\left (\frac {1}{z}\right )^{\text {n3}-1}}{\text {n3}-1}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=   a__1*x^(n__1)*diff(w(x,y,z),x)+ a__2*y^(n__2)*diff(w(x,y,z),y)+ a__3*z^(n__3)*diff(w(x,y,z),z)=b*x^k*w(x,y,z)+ x*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {6 \left (-\frac {\left (-4 \left (-\frac {k}{2}-\frac {m}{2}+n_{1} -\frac {3}{2}\right )^{2} a_{1} \WhittakerM \left (\frac {k +m -2 n_{1} +3}{2 k -2 n_{1} +2}, \frac {2 k +m -3 n_{1} +4}{2 k -2 n_{1} +2}, \frac {b \,x^{k -n_{1} +1}}{\left (k -n_{1} +1\right ) a_{1}}\right )+\left (b \,x^{k -n_{1} +1}-2 \left (-\frac {k}{2}-\frac {m}{2}+n_{1} -\frac {3}{2}\right ) a_{1} \right ) \left (-k +n_{1} -1\right ) \WhittakerM \left (\frac {-k +m +1}{2 k -2 n_{1} +2}, \frac {2 k +m -3 n_{1} +4}{2 k -2 n_{1} +2}, \frac {b \,x^{k -n_{1} +1}}{\left (k -n_{1} +1\right ) a_{1}}\right )\right ) \left (-k +n_{1} -1\right ) x^{-k +m +1} \left (\frac {b}{\left (k -n_{1} +1\right ) a_{1}}\right )^{\frac {-m +n_{1} -2}{k -n_{1} +1}} \left (\frac {b}{\left (k -n_{1} +1\right ) a_{1}}\right )^{\frac {m -n_{1} +2}{k -n_{1} +1}} \left (\frac {b \,x^{k -n_{1} +1}}{\left (k -n_{1} +1\right ) a_{1}}\right )^{\frac {-k -m +2 n_{1} -3}{2 k -2 n_{1} +2}} {\mathrm e}^{-\frac {b \,x^{k -n_{1} +1}}{2 \left (k -n_{1} +1\right ) a_{1}}}}{6}+\left (-\frac {k}{2}-\frac {m}{2}+n_{1} -\frac {3}{2}\right ) \left (-m +n_{1} -2\right ) \left (-\frac {2 k}{3}-\frac {m}{3}+n_{1} -\frac {4}{3}\right ) a_{1} b \textit {\_F1} \left (\frac {\left (n_{1} -1\right ) a_{1} y^{-n_{2} +1}-\left (n_{2} -1\right ) a_{2} x^{-n_{1} +1}}{\left (n_{1} -1\right ) a_{1}}, \frac {\left (n_{1} -1\right ) a_{1} z^{-n_{3} +1}-\left (n_{3} -1\right ) a_{3} x^{-n_{1} +1}}{\left (n_{1} -1\right ) a_{1}}\right )\right ) {\mathrm e}^{\frac {b \,x^{k -n_{1} +1}}{\left (k -n_{1} +1\right ) a_{1}}}}{\left (-m +n_{1} -2\right ) \left (-k -m +2 n_{1} -3\right ) \left (-2 k -m +3 n_{1} -4\right ) a_{1} b}\]

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