#### 6.9.4 2.4

6.9.4.1 [1941] Problem 1
6.9.4.2 [1942] Problem 2
6.9.4.3 [1943] Problem 3
6.9.4.4 [1944] Problem 4
6.9.4.5 [1945] Problem 5
6.9.4.6 [1946] Problem 6
6.9.4.7 [1947] Problem 7
6.9.4.8 [1948] Problem 8
6.9.4.9 [1949] Problem 9
6.9.4.10 [1950] Problem 10
6.9.4.11 [1951] Problem 11
6.9.4.12 [1952] Problem 12
6.9.4.13 [1953] Problem 13
6.9.4.14 [1954] Problem 14

##### 6.9.4.1 [1941] Problem 1

problem number 1941

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]];



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

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'));



$w \left ( x,y,z \right ) ={\frac {1}{ak \left ( m+2\,n+3 \right ) \left ( m+n+2 \right ) \left ( m+1 \right ) }{{\rm e}^{{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}}}} \left ( \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-1}{n+1}}}s \left ( {x}^{n+1}k+a \left ( m+n+2 \right ) \right ) \left ( {\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-n-2}{2\,n+2}}} \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {m+1}{n+1}}}{{\rm e}^{-1/2\,{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}}}} \left ( n+1 \right ) ^{2}{x}^{-n+m} \WhittakerM \left ( {\frac {-n+m}{2\,n+2}},{\frac {m+2\,n+3}{2\,n+2}},{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) + \left ( {{\rm e}^{-1/2\,{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}}}} \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-1}{n+1}}} \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {m+1}{n+1}}} \left ( {\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-n-2}{2\,n+2}}}{x}^{-n+m}s \left ( n+1 \right ) \left ( m+n+2 \right ) \WhittakerM \left ( {\frac {m+n+2}{2\,n+2}},{\frac {m+2\,n+3}{2\,n+2}},{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) +{\it \_F1} \left ( {\frac {ay-bx}{a}},{\frac {za-cx}{a}} \right ) k \left ( m+1 \right ) \left ( m+2\,n+3 \right ) \right ) a \left ( m+n+2 \right ) \right ) }$

____________________________________________________________________________________

##### 6.9.4.2 [1942] Problem 2

problem number 1942

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]];



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

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'));



$w \left ( x,y,z \right ) ={\frac {1}{ak \left ( m+2\,n+3 \right ) \left ( m+n+2 \right ) \left ( m+1 \right ) }{{\rm e}^{{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}}}} \left ( \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-1}{n+1}}}s \left ( {x}^{n+1}k+a \left ( m+n+2 \right ) \right ) \left ( {\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-n-2}{2\,n+2}}} \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {m+1}{n+1}}}{{\rm e}^{-1/2\,{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}}}} \left ( n+1 \right ) ^{2}{x}^{-n+m} \WhittakerM \left ( {\frac {-n+m}{2\,n+2}},{\frac {m+2\,n+3}{2\,n+2}},{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) +a \left ( {{\rm e}^{-1/2\,{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}}}} \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-1}{n+1}}} \left ( {\frac {k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {m+1}{n+1}}} \left ( {\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) ^{{\frac {-m-n-2}{2\,n+2}}}{x}^{-n+m}s \left ( n+1 \right ) \left ( m+n+2 \right ) \WhittakerM \left ( {\frac {m+n+2}{2\,n+2}},{\frac {m+2\,n+3}{2\,n+2}},{\frac {{x}^{n+1}k}{ \left ( n+1 \right ) a}} \right ) +{\it \_F1} \left ( y{{\rm e}^{-{\frac {bx}{a}}}},z{{\rm e}^{-{\frac {cx}{a}}}} \right ) k \left ( m+1 \right ) \left ( m+2\,n+3 \right ) \right ) \left ( m+n+2 \right ) \right ) }$

____________________________________________________________________________________

##### 6.9.4.3 [1943] Problem 3

problem number 1943

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]];



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

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'));



$w \left ( x,y,z \right ) = \left ( \int ^{y}\!{\frac {s}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }} \left ( {\frac {1}{\sqrt {ab}} \left ( x\sqrt {ab}-\ln \left ( {\frac {aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab}}{\sqrt {ab}}} \right ) +\ln \left ( {\frac {ab{\it \_b}+\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) \right ) } \right ) ^{m}{{\rm e}^{-c\int \!{\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }} \left ( {\frac {1}{\sqrt {ab}} \left ( x\sqrt {ab}-\ln \left ( {\frac {aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab}}{\sqrt {ab}}} \right ) +\ln \left ( {\frac {ab{\it \_b}+\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) \right ) } \right ) ^{n}}\,{\rm d}{\it \_b}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {a{z}^{2}-b{y}^{2}}{a}},-{\frac {1}{\sqrt {ab}} \left ( -x\sqrt {ab}+\ln \left ( {\frac {aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab}}{\sqrt {ab}}} \right ) \right ) } \right ) \right ) {{\rm e}^{\int ^{y}\!{\frac {c}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) }} \left ( {\frac {1}{\sqrt {ab}} \left ( x\sqrt {ab}-\ln \left ( {\frac {aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab}}{\sqrt {ab}}} \right ) +\ln \left ( {\frac {ab{\it \_a}+\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) \right ) } \right ) ^{n}}{d{\it \_a}}}}$

____________________________________________________________________________________

##### 6.9.4.4 [1944] Problem 4

problem number 1944

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]];



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

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'));



$w \left ( x,y,z \right ) ={\frac {1}{c \left ( 2\,k+r+3 \right ) \left ( r+1 \right ) \left ( k+r+2 \right ) } \left ( 2\, \left ( k+r/2+3/2 \right ) \left ( r+1 \right ) c \left ( k+r+2 \right ) {\it \_F1} \left ( {\frac {-xa{x}^{n}+y \left ( n+1 \right ) }{n+1}},{\frac {-xb{x}^{m}+z \left ( m+1 \right ) }{m+1}} \right ) +{{\rm e}^{-{\frac {{x}^{k+1}c}{2\,k+2}}}}{x}^{-k+r} \left ( k+1 \right ) \left ( {\frac {{x}^{k+1}c}{k+1}} \right ) ^{{\frac {-k-r-2}{2\,k+2}}}s \left ( {\frac {c}{k+1}} \right ) ^{{\frac {-r-1}{k+1}}} \left ( \left ( k+1 \right ) \left ( {x}^{k+1}c+k+r+2 \right ) \WhittakerM \left ( {\frac {-k+r}{2\,k+2}},{\frac {2\,k+r+3}{2\,k+2}},{\frac {{x}^{k+1}c}{k+1}} \right ) + \WhittakerM \left ( {\frac {k+r+2}{2\,k+2}},{\frac {2\,k+r+3}{2\,k+2}},{\frac {{x}^{k+1}c}{k+1}} \right ) \left ( k+r+2 \right ) ^{2} \right ) \left ( {\frac {c}{k+1}} \right ) ^{{\frac {r+1}{k+1}}} \right ) {{\rm e}^{{\frac {{x}^{k+1}c}{k+1}}}}}$

____________________________________________________________________________________

##### 6.9.4.5 [1945] Problem 5

problem number 1945

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]];



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

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'));



$w \left ( x,y,z \right ) ={\frac {{{\rm e}^{ax}}}{a \left ( k+1 \right ) } \left ( a \left ( k+1 \right ) {\it \_F1} \left ( {\frac {-xb{x}^{n}+y \left ( n+1 \right ) }{n+1}},{\frac {-xc{x}^{m}+z \left ( m+1 \right ) }{m+1}} \right ) +s{x}^{k} \left ( ax \right ) ^{-k/2}{{\rm e}^{-1/2\,ax}} \WhittakerM \left ( k/2,k/2+1/2,ax \right ) \right ) }$

____________________________________________________________________________________

##### 6.9.4.6 [1946] Problem 6

problem number 1946

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]];



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

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'));



$w \left ( x,y,z \right ) ={\frac {1}{c \left ( 2\,k+r+3 \right ) \left ( r+1 \right ) \left ( k+r+2 \right ) } \left ( 2\, \left ( k+r/2+3/2 \right ) \left ( r+1 \right ) c \left ( k+r+2 \right ) {\it \_F1} \left ( {\frac { \left ( -{x}^{n} \left ( ax \right ) ^{-n/2} \WhittakerM \left ( n/2,n/2+1/2,ax \right ) {{\rm e}^{1/2\,ax}}\beta +ay \left ( n+1 \right ) \right ) {{\rm e}^{-ax}}}{ \left ( n+1 \right ) a}},{\frac { \left ( -{x}^{m}{{\rm e}^{1/2\,bx}} \left ( bx \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,bx \right ) \gamma +bz \left ( m+1 \right ) \right ) {{\rm e}^{-bx}}}{b \left ( m+1 \right ) }} \right ) +{{\rm e}^{-{\frac {{x}^{k+1}c}{2\,k+2}}}}{x}^{-k+r} \left ( k+1 \right ) \left ( {\frac {{x}^{k+1}c}{k+1}} \right ) ^{{\frac {-k-r-2}{2\,k+2}}}s \left ( {\frac {c}{k+1}} \right ) ^{{\frac {-r-1}{k+1}}} \left ( \left ( k+1 \right ) \left ( {x}^{k+1}c+k+r+2 \right ) \WhittakerM \left ( {\frac {-k+r}{2\,k+2}},{\frac {2\,k+r+3}{2\,k+2}},{\frac {{x}^{k+1}c}{k+1}} \right ) + \WhittakerM \left ( {\frac {k+r+2}{2\,k+2}},{\frac {2\,k+r+3}{2\,k+2}},{\frac {{x}^{k+1}c}{k+1}} \right ) \left ( k+r+2 \right ) ^{2} \right ) \left ( {\frac {c}{k+1}} \right ) ^{{\frac {r+1}{k+1}}} \right ) {{\rm e}^{{\frac {{x}^{k+1}c}{k+1}}}}}$

____________________________________________________________________________________

##### 6.9.4.7 [1947] Problem 7

problem number 1947

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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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'));



$\text {Expression too large to display}$

____________________________________________________________________________________

##### 6.9.4.8 [1948] Problem 8

problem number 1948

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}} \text {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}} \text {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}} \text {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}} \text {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 {1}{a_{1}\, \left ( 2\,\lambda _{1}+\lambda _{2}+3 \right ) \left ( \lambda _{1}+\lambda _{2}+2 \right ) \left ( \lambda _{2}+1 \right ) } \left ( \left ( \left ( -\lambda _{1}-\lambda _{2}-2 \right ) {x}^{\lambda _{2}-\lambda _{1}}+a_{1}\,{x}^{\lambda _{2}+1} \left ( k-1 \right ) \right ) a_{2}\,{y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}}{y}^{{\frac {k}{\lambda _{1}+1}}}{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{2\,\lambda _{1}+2}}}}{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \left ( \lambda _{1}+1 \right ) ^{2} \left ( -{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) ^{{\frac {-\lambda _{1}-\lambda _{2}-2}{2\,\lambda _{1}+2}}} \WhittakerM \left ( {\frac {\lambda _{2}-\lambda _{1}}{2\,\lambda _{1}+2}},{\frac {2\,\lambda _{1}+\lambda _{2}+3}{2\,\lambda _{1}+2}},-{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) - \left ( \left ( -\lambda _{1}-\lambda _{2}-2 \right ) {{\it \_a}}^{\lambda _{2}-\lambda _{1}}+{{\it \_a}}^{\lambda _{2}+1}a_{1}\, \left ( k-1 \right ) \right ) \left ( \lambda _{1}+1 \right ) ^{2}{{\rm e}^{{\frac {a_{1}\,{{\it \_a}}^{\lambda _{1}+1} \left ( k-1 \right ) }{2\,\lambda _{1}+2}}}}{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \left ( -{\frac {a_{1}\,{{\it \_a}}^{\lambda _{1}+1} \left ( k-1 \right ) }{\lambda _{1}+1}} \right ) ^{{\frac {-\lambda _{1}-\lambda _{2}-2}{2\,\lambda _{1}+2}}}a_{2}\,{y}^{{\frac {k}{\lambda _{1}+1}}}{y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}} \WhittakerM \left ( {\frac {\lambda _{2}-\lambda _{1}}{2\,\lambda _{1}+2}},{\frac {2\,\lambda _{1}+\lambda _{2}+3}{2\,\lambda _{1}+2}},-{\frac {a_{1}\,{{\it \_a}}^{\lambda _{1}+1} \left ( k-1 \right ) }{\lambda _{1}+1}} \right ) +2\, \left ( -1/2\,{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{2\,\lambda _{1}+2}}}}{x}^{\lambda _{2}-\lambda _{1}} \left ( -{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) ^{{\frac {-\lambda _{1}-\lambda _{2}-2}{2\,\lambda _{1}+2}}}a_{2}\,{y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}}{y}^{{\frac {k}{\lambda _{1}+1}}}{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \left ( \lambda _{1}+1 \right ) \left ( \lambda _{1}+\lambda _{2}+2 \right ) \WhittakerM \left ( {\frac {\lambda _{1}+\lambda _{2}+2}{2\,\lambda _{1}+2}},{\frac {2\,\lambda _{1}+\lambda _{2}+3}{2\,\lambda _{1}+2}},-{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) +1/2\,{{\rm e}^{{\frac {a_{1}\,{{\it \_a}}^{\lambda _{1}+1} \left ( k-1 \right ) }{2\,\lambda _{1}+2}}}}{{\it \_a}}^{\lambda _{2}-\lambda _{1}} \left ( -{\frac {a_{1}\,{{\it \_a}}^{\lambda _{1}+1} \left ( k-1 \right ) }{\lambda _{1}+1}} \right ) ^{{\frac {-\lambda _{1}-\lambda _{2}-2}{2\,\lambda _{1}+2}}}a_{2}\,{y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}}{y}^{{\frac {k}{\lambda _{1}+1}}}{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \left ( \lambda _{1}+1 \right ) \left ( \lambda _{1}+\lambda _{2}+2 \right ) \WhittakerM \left ( {\frac {\lambda _{1}+\lambda _{2}+2}{2\,\lambda _{1}+2}},{\frac {2\,\lambda _{1}+\lambda _{2}+3}{2\,\lambda _{1}+2}},-{\frac {a_{1}\,{{\it \_a}}^{\lambda _{1}+1} \left ( k-1 \right ) }{\lambda _{1}+1}} \right ) +{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}k}{\lambda _{1}+1}}}} \left ( \lambda _{2}+1 \right ) {y}^{ \left ( \lambda _{1}+1 \right ) ^{-1}}a_{1}\, \left ( \lambda _{1}+\lambda _{2}/2+3/2 \right ) {y}^{{\frac {\lambda _{1}}{\lambda _{1}+1}}} \right ) \left ( \lambda _{1}+\lambda _{2}+2 \right ) \right ) \left ( {{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \right ) ^{-1} \left ( {y}^{{\frac {k}{\lambda _{1}+1}}} \right ) ^{-1} \left ( {y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{{\frac {a_{1}\,{{\it \_a}}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \right ) ^{\gamma _{2}}{{\rm e}^{-{\frac {c_{1}\,{{\it \_a}}^{\gamma _{1}+1}}{\gamma _{1}+1}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {1}{a_{1}\, \left ( 2\,\lambda _{1}+\lambda _{2}+3 \right ) \left ( \lambda _{1}+\lambda _{2}+2 \right ) \left ( \lambda _{2}+1 \right ) } \left ( \left ( \left ( -\lambda _{1}-\lambda _{2}-2 \right ) {x}^{\lambda _{2}-\lambda _{1}}+a_{1}\,{x}^{\lambda _{2}+1} \left ( k-1 \right ) \right ) a_{2}\,{y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}}{y}^{{\frac {k}{\lambda _{1}+1}}}{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{2\,\lambda _{1}+2}}}}{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \left ( \lambda _{1}+1 \right ) ^{2} \left ( -{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) ^{{\frac {-\lambda _{1}-\lambda _{2}-2}{2\,\lambda _{1}+2}}} \WhittakerM \left ( {\frac {\lambda _{2}-\lambda _{1}}{2\,\lambda _{1}+2}},{\frac {2\,\lambda _{1}+\lambda _{2}+3}{2\,\lambda _{1}+2}},-{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) +2\, \left ( \lambda _{1}+\lambda _{2}+2 \right ) \left ( -1/2\,{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{2\,\lambda _{1}+2}}}}{x}^{\lambda _{2}-\lambda _{1}} \left ( -{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) ^{{\frac {-\lambda _{1}-\lambda _{2}-2}{2\,\lambda _{1}+2}}}a_{2}\,{y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}}{y}^{{\frac {k}{\lambda _{1}+1}}}{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \left ( \lambda _{1}+1 \right ) \left ( \lambda _{1}+\lambda _{2}+2 \right ) \WhittakerM \left ( {\frac {\lambda _{1}+\lambda _{2}+2}{2\,\lambda _{1}+2}},{\frac {2\,\lambda _{1}+\lambda _{2}+3}{2\,\lambda _{1}+2}},-{\frac {a_{1}\, \left ( k-1 \right ) {x}^{\lambda _{1}+1}}{\lambda _{1}+1}} \right ) +{{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}k}{\lambda _{1}+1}}}} \left ( \lambda _{2}+1 \right ) {y}^{ \left ( \lambda _{1}+1 \right ) ^{-1}}a_{1}\, \left ( \lambda _{1}+\lambda _{2}/2+3/2 \right ) {y}^{{\frac {\lambda _{1}}{\lambda _{1}+1}}} \right ) \right ) \left ( {{\rm e}^{{\frac {a_{1}\,{x}^{\lambda _{1}+1}}{\lambda _{1}+1}}}} \right ) ^{-1} \left ( {y}^{{\frac {k}{\lambda _{1}+1}}} \right ) ^{-1} \left ( {y}^{{\frac {k\lambda _{1}}{\lambda _{1}+1}}} \right ) ^{-1}},{\frac {1}{b_{1}\, \left ( \beta _{2}+1 \right ) \left ( \beta _{1}+\beta _{2}+2 \right ) \left ( 2\,\beta _{1}+\beta _{2}+3 \right ) } \left ( \left ( \left ( -\beta _{1}-\beta _{2}-2 \right ) {x}^{-\beta _{1}+\beta _{2}}+b_{1}\,{x}^{\beta _{2}+1} \left ( m-1 \right ) \right ) b_{2}\,{z}^{{\frac {\beta _{1}\,m}{\beta _{1}+1}}}{z}^{{\frac {m}{\beta _{1}+1}}}{{\rm e}^{{\frac {{x}^{\beta _{1}+1}b_{1}\, \left ( m-1 \right ) }{2\,\beta _{1}+2}}}}{{\rm e}^{{\frac {b_{1}\,{x}^{\beta _{1}+1}}{\beta _{1}+1}}}} \left ( \beta _{1}+1 \right ) ^{2} \left ( -{\frac {{x}^{\beta _{1}+1}b_{1}\, \left ( m-1 \right ) }{\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 {{x}^{\beta _{1}+1}b_{1}\, \left ( m-1 \right ) }{\beta _{1}+1}} \right ) +2\, \left ( \beta _{1}+\beta _{2}+2 \right ) \left ( -1/2\,{{\rm e}^{{\frac {{x}^{\beta _{1}+1}b_{1}\, \left ( m-1 \right ) }{2\,\beta _{1}+2}}}}{x}^{-\beta _{1}+\beta _{2}} \left ( -{\frac {{x}^{\beta _{1}+1}b_{1}\, \left ( m-1 \right ) }{\beta _{1}+1}} \right ) ^{{\frac {-\beta _{1}-\beta _{2}-2}{2\,\beta _{1}+2}}}b_{2}\,{z}^{{\frac {\beta _{1}\,m}{\beta _{1}+1}}}{z}^{{\frac {m}{\beta _{1}+1}}}{{\rm e}^{{\frac {b_{1}\,{x}^{\beta _{1}+1}}{\beta _{1}+1}}}} \left ( \beta _{1}+1 \right ) \left ( \beta _{1}+\beta _{2}+2 \right ) \WhittakerM \left ( {\frac {\beta _{1}+\beta _{2}+2}{2\,\beta _{1}+2}},{\frac {2\,\beta _{1}+\beta _{2}+3}{2\,\beta _{1}+2}},-{\frac {{x}^{\beta _{1}+1}b_{1}\, \left ( m-1 \right ) }{\beta _{1}+1}} \right ) +{{\rm e}^{{\frac {b_{1}\,{x}^{\beta _{1}+1}m}{\beta _{1}+1}}}} \left ( \beta _{2}+1 \right ) {z}^{ \left ( \beta _{1}+1 \right ) ^{-1}}b_{1}\, \left ( \beta _{1}+\beta _{2}/2+3/2 \right ) {z}^{{\frac {\beta _{1}}{\beta _{1}+1}}} \right ) \right ) \left ( {z}^{{\frac {\beta _{1}\,m}{\beta _{1}+1}}} \right ) ^{-1} \left ( {z}^{{\frac {m}{\beta _{1}+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b_{1}\,{x}^{\beta _{1}+1}}{\beta _{1}+1}}}} \right ) ^{-1}} \right ) \right ) {{\rm e}^{{\frac {{x}^{\gamma _{1}+1}c_{1}}{\gamma _{1}+1}}}}$

____________________________________________________________________________________

##### 6.9.4.9 [1949] Problem 9

problem number 1949

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'));



time expired

____________________________________________________________________________________

##### 6.9.4.10 [1950] Problem 10

problem number 1950

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]];



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

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'));



$w \left ( x,y,z \right ) ={\frac {1}{m \left ( m+n \right ) \left ( m+2\,n \right ) c}{{\rm e}^{{\frac {{x}^{n}c}{n}}}} \left ( {x}^{-n+m} \left ( {\frac {c}{n}} \right ) ^{{\frac {m}{n}}} \left ( {\frac {c}{n}} \right ) ^{-{\frac {m}{n}}}{{\rm e}^{-1/2\,{\frac {{x}^{n}c}{n}}}} \left ( {\frac {{x}^{n}c}{n}} \right ) ^{-1/2\,{\frac {m+n}{n}}}k{n}^{2} \left ( {x}^{n}c+m+n \right ) \WhittakerM \left ( 1/2\,{\frac {-n+m}{n}},1/2\,{\frac {m+2\,n}{n}},{\frac {{x}^{n}c}{n}} \right ) + \left ( {x}^{-n+m} \left ( {\frac {c}{n}} \right ) ^{{\frac {m}{n}}} \left ( {\frac {c}{n}} \right ) ^{-{\frac {m}{n}}}{{\rm e}^{-1/2\,{\frac {{x}^{n}c}{n}}}} \left ( {\frac {{x}^{n}c}{n}} \right ) ^{-1/2\,{\frac {m+n}{n}}}kn \left ( m+n \right ) \WhittakerM \left ( 1/2\,{\frac {m+n}{n}},1/2\,{\frac {m+2\,n}{n}},{\frac {{x}^{n}c}{n}} \right ) +{\it \_F1} \left ( y{x}^{-a},z{x}^{-b} \right ) cm \left ( m+2\,n \right ) \right ) \left ( m+n \right ) \right ) }$

____________________________________________________________________________________

##### 6.9.4.11 [1951] Problem 11

problem number 1951

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]];



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

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'));



$w \left ( x,y,z \right ) = \left ( \int ^{y}\!{\frac {k}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }} \left ( x \left ( \sqrt {ab}y+az \right ) ^{-{\frac {\sqrt {ab}}{ab}}} \left ( {\frac {ab{\it \_b}+\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) ^{{\frac {1}{\sqrt {ab}}}} \right ) ^{m}{{\rm e}^{-c\int \!{\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }} \left ( x \left ( \sqrt {ab}y+az \right ) ^{-{\frac {\sqrt {ab}}{ab}}} \left ( {\frac {ab{\it \_b}+\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) ^{{\frac {1}{\sqrt {ab}}}} \right ) ^{n}}\,{\rm d}{\it \_b}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {a{z}^{2}-b{y}^{2}}{a}},x \left ( \sqrt {ab}y+az \right ) ^{-{\frac {\sqrt {ab}}{ab}}} \right ) \right ) {{\rm e}^{\int ^{y}\!{\frac {c}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) }} \left ( x \left ( \sqrt {ab}y+az \right ) ^{-{\frac {\sqrt {ab}}{ab}}} \left ( {\frac {ab{\it \_a}+\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) ^{{\frac {1}{\sqrt {ab}}}} \right ) ^{n}}{d{\it \_a}}}}$

____________________________________________________________________________________

##### 6.9.4.12 [1952] Problem 12

problem number 1952

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 ( -\int ^{y}\!-{\frac {s}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}c} \left ( x \left ( \left ( {\frac {\sqrt {2}{b}^{2}y}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+ \left ( {\frac {by}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}+{\frac {cz}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) \sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}} \right ) ^{-1/2\,{\frac {b\sqrt {2}}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}}}} \left ( \left ( {\frac {{b}^{2}{\it \_b}\,\sqrt {2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+\sqrt {2\,{\frac {{b}^{2}{{\it \_b}}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+1}\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}} \right ) ^{1/2\,{\frac {b\sqrt {2}}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}}}} \right ) ^{m}{{\rm e}^{-{\frac {k}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}c}\int \! \left ( x \left ( \left ( {\frac {\sqrt {2}{b}^{2}y}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+ \left ( {\frac {by}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}+{\frac {cz}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) \sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}} \right ) ^{-1/2\,{\frac {b\sqrt {2}}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}}}} \left ( \left ( {\frac {{b}^{2}{\it \_b}\,\sqrt {2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+\sqrt {2\,{\frac {{b}^{2}{{\it \_b}}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+1}\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}} \right ) ^{1/2\,{\frac {b\sqrt {2}}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}}}} \right ) ^{n}{\frac {1}{\sqrt {2\,{\frac {{b}^{2}{{\it \_b}}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+1}}}\,{\rm d}{\it \_b}}}}{\frac {1}{\sqrt {2\,{\frac {{b}^{2}{{\it \_b}}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+1}}}}{d{\it \_b}}+{\it \_F1} \left ( -{\frac {1}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}},x \left ( \left ( {\frac {\sqrt {2}{b}^{2}y}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+ \left ( {\frac {by}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}+{\frac {cz}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) \sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}} \right ) ^{-1/2\,{\frac {b\sqrt {2}}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}}}} \right ) \right ) {{\rm e}^{-\int ^{y}\!-{\frac {k}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}c} \left ( x \left ( \left ( {\frac {\sqrt {2}{b}^{2}y}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+ \left ( {\frac {by}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}+{\frac {cz}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) \sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}} \right ) ^{-1/2\,{\frac {b\sqrt {2}}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}}}} \left ( \left ( {\frac {{\it \_a}\,{b}^{2}\sqrt {2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+\sqrt {2\,{\frac {{{\it \_a}}^{2}{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+1}\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}} \right ) ^{1/2\,{\frac {b\sqrt {2}}{\sqrt {-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}}}}}} \right ) ^{n}{\frac {1}{\sqrt {2\,{\frac {{{\it \_a}}^{2}{b}^{2}}{-{b}^{2}{y}^{2}+2\,bzcy+{c}^{2}{z}^{2}}}+1}}}}{d{\it \_a}}}}$

____________________________________________________________________________________

##### 6.9.4.13 [1953] Problem 13

problem number 1953

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'));



$w \left ( x,y,z \right ) = \left ( \int ^{x}\!{\frac {1}{b_{1}}{{\rm e}^{{\frac {{{\it \_a}}^{-n_{1}+1}a}{ \left ( n_{1}-1 \right ) b_{1}}}}} \left ( {{\it \_a}}^{-n_{1}}c_{2}\, \left ( \left ( {\frac {-{x}^{-n_{1}+1}b_{2}\, \left ( n_{2}-1 \right ) +{y}^{1-n_{2}}b_{1}\, \left ( n_{1}-1 \right ) +{{\it \_a}}^{-n_{1}+1}b_{2}\, \left ( n_{2}-1 \right ) }{ \left ( n_{1}-1 \right ) b_{1}}} \right ) ^{- \left ( n_{2}-1 \right ) ^{-1}} \right ) ^{k_{2}}+{{\it \_a}}^{-n_{1}+k_{1}}c_{1}+{{\it \_a}}^{-n_{1}+k_{3}}c_{3} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {-{x}^{-n_{1}+1}b_{2}\, \left ( n_{2}-1 \right ) +{y}^{1-n_{2}}b_{1}\, \left ( n_{1}-1 \right ) }{ \left ( n_{1}-1 \right ) b_{1}}},{\frac {-{x}^{-n_{1}+1}b_{3}\, \left ( n_{3}-1 \right ) +{z}^{1-n_{3}}b_{1}\, \left ( n_{1}-1 \right ) }{ \left ( n_{1}-1 \right ) b_{1}}} \right ) \right ) {{\rm e}^{-{\frac {{x}^{-n_{1}+1}a}{ \left ( n_{1}-1 \right ) b_{1}}}}}$

____________________________________________________________________________________

##### 6.9.4.14 [1954] Problem 14

problem number 1954

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]];



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

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'));



$w \left ( x,y,z \right ) =6\,{\frac {1}{ \left ( -m+n_{1}-2 \right ) a_{1}\, \left ( -k-m+2\,n_{1}-3 \right ) \left ( -2\,k-m+3\,n_{1}-4 \right ) b}{{\rm e}^{{\frac {b{x}^{k-n_{1}+1}}{a_{1}\, \left ( k-n_{1}+1 \right ) }}}} \left ( a_{1}\, \left ( -m+n_{1}-2 \right ) b \left ( -k/2-m/2+n_{1}-3/2 \right ) \left ( -2/3\,k-m/3+n_{1}-4/3 \right ) {\it \_F1} \left ( {\frac {-{x}^{-n_{1}+1}a_{2}\, \left ( n_{2}-1 \right ) +{y}^{1-n_{2}}a_{1}\, \left ( n_{1}-1 \right ) }{ \left ( n_{1}-1 \right ) a_{1}}},{\frac {-{x}^{-n_{1}+1}a_{3}\, \left ( n_{3}-1 \right ) +{z}^{1-n_{3}}a_{1}\, \left ( n_{1}-1 \right ) }{ \left ( n_{1}-1 \right ) a_{1}}} \right ) -1/6\, \left ( {\frac {b}{a_{1}\, \left ( k-n_{1}+1 \right ) }} \right ) ^{{\frac {m-n_{1}+2}{k-n_{1}+1}}} \left ( {\frac {b{x}^{k-n_{1}+1}}{a_{1}\, \left ( k-n_{1}+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n_{1}-3}{2\,k-2\,n_{1}+2}}} \left ( {\frac {b}{a_{1}\, \left ( k-n_{1}+1 \right ) }} \right ) ^{{\frac {-m+n_{1}-2}{k-n_{1}+1}}} \left ( -4\,a_{1}\, \left ( -k/2-m/2+n_{1}-3/2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n_{1}+3}{2\,k-2\,n_{1}+2}},{\frac {2\,k+m-3\,n_{1}+4}{2\,k-2\,n_{1}+2}},{\frac {b{x}^{k-n_{1}+1}}{a_{1}\, \left ( k-n_{1}+1 \right ) }} \right ) + \left ( b{x}^{k-n_{1}+1}-2\,a_{1}\, \left ( -k/2-m/2+n_{1}-3/2 \right ) \right ) \WhittakerM \left ( {\frac {-k+1+m}{2\,k-2\,n_{1}+2}},{\frac {2\,k+m-3\,n_{1}+4}{2\,k-2\,n_{1}+2}},{\frac {b{x}^{k-n_{1}+1}}{a_{1}\, \left ( k-n_{1}+1 \right ) }} \right ) \left ( -k+n_{1}-1 \right ) \right ) {{\rm e}^{-1/2\,{\frac {b{x}^{k-n_{1}+1}}{a_{1}\, \left ( k-n_{1}+1 \right ) }}}}{x}^{-k+1+m} \left ( -k+n_{1}-1 \right ) \right ) }$

____________________________________________________________________________________