#### 6.9.1 2.1

6.9.1.1 [1920] Problem 1
6.9.1.2 [1921] Problem 2
6.9.1.3 [1922] Problem 3
6.9.1.4 [1923] Problem 4
6.9.1.5 [1924] Problem 5
6.9.1.6 [1925] Problem 6
6.9.1.7 [1926] Problem 7
6.9.1.8 [1927] Problem 8
6.9.1.9 [1928] Problem 9

##### 6.9.1.1 [1920] Problem 1

problem number 1920

Added Jan 6, 2020.

Problem Chapter 9.2.1.1, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $a w_x + b w_y + c w_z = (\alpha x + \beta ) w + p x + q$

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]==(alpha*x+beta)*w[x,y,z]+p*x+q;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to e^{\frac {x (\alpha x+2 \beta )}{2 a}} c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\alpha x+\beta )^2}{2 a \alpha }} (\alpha q-\beta p) \text {Erf}\left (\frac {\alpha x+\beta }{\sqrt {2} \sqrt {a} \sqrt {\alpha }}\right )}{\sqrt {a} \alpha ^{3/2}}-\frac {p}{\alpha }\right \}\right \}$

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)=(alpha*x+beta)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



$w \left ( x,y,z \right ) ={\frac {1}{a\alpha } \left ( -1/2\,\sqrt {\pi }{{\rm e}^{1/2\,{\frac {{\beta }^{2}}{a\alpha }}}}\sqrt {2} \left ( -\alpha \,q+\beta \,p \right ) \erf \left ( 1/2\,\sqrt {2}\sqrt {{\frac {\alpha }{a}}}x+1/2\,{\frac {\beta \,\sqrt {2}}{a}{\frac {1}{\sqrt {{\frac {\alpha }{a}}}}}} \right ) +a\sqrt {{\frac {\alpha }{a}}} \left ( \alpha \,{\it \_F1} \left ( {\frac {ay-bx}{a}},{\frac {za-cx}{a}} \right ) -{{\rm e}^{-1/2\,{\frac {x \left ( \alpha \,x+2\,\beta \right ) }{a}}}}p \right ) \right ) {{\rm e}^{1/2\,{\frac {x \left ( \alpha \,x+2\,\beta \right ) }{a}}}}{\frac {1}{\sqrt {{\frac {\alpha }{a}}}}}}$

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##### 6.9.1.2 [1921] Problem 2

problem number 1921

Added Jan 6, 2020.

Problem Chapter 9.2.1.2, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + a z w_y + b y w_z = (c x + k) w + p x + q$

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+k)*w[x,y,z]+p*x+q;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (c x+2 k)} c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(c x+k)^2}{2 c}} \text {Erf}\left (\frac {c x+k}{\sqrt {2} \sqrt {c}}\right ) (c q-k p)}{c^{3/2}}-\frac {p}{c}\right \}\right \}$

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+k)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



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

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##### 6.9.1.3 [1922] Problem 3

problem number 1922

Added Jan 6, 2020.

Problem Chapter 9.2.1.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + (a_1 x+a_0) w_y + (b_1 x+b_0) w_z = (c_1 x + c_0) w + s_1 x + s_0$

Mathematica

ClearAll["Global*"];
pde =  D[w[x,y,z],x]+(a1*x+a0)*D[w[x,y,z],y]+(b1*x+b0)*D[w[x,y,z],z]==(c1*x+c0)*w[x,y,z]+s1*x+s0;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (-\text {a0} x-\frac {\text {a1} x^2}{2}+y,-\text {b0} x-\frac {\text {b1} x^2}{2}+z\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {Erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}$

Maple

restart;
local gamma;
pde :=  diff(w(x,y,z),x)+ (a__1*x+a__0)*diff(w(x,y,z),y)+ (b__1*x+b__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



$w \left ( x,y,z \right ) =1/2\,{\frac {{{\rm e}^{1/2\,x \left ( c_{1}\,x+2\,c_{0} \right ) }}}{{c_{1}}^{5/2}} \left ( 2\,{\it \_F1} \left ( -1/2\,a_{1}\,{x}^{2}-a_{0}\,x+y,-1/2\,b_{1}\,{x}^{2}-b_{0}\,x+z \right ) {c_{1}}^{5/2}+c_{1}\,\sqrt {2}\sqrt {\pi }{{\rm e}^{1/2\,{\frac {{c_{0}}^{2}}{c_{1}}}}} \left ( -c_{0}\,s_{1}+c_{1}\,s_{0} \right ) \erf \left ( 1/2\,\sqrt {2} \left ( x\sqrt {c_{1}}+{\frac {c_{0}}{\sqrt {c_{1}}}} \right ) \right ) -2\,{{\rm e}^{-1/2\,c_{1}\,{x}^{2}-c_{0}\,x}}{c_{1}}^{3/2}s_{1} \right ) }$

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##### 6.9.1.4 [1923] Problem 4

problem number 1923

Added Jan 6, 2020.

Problem Chapter 9.2.1.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + (b_1 x+b_0) w_y + (c_1 y+c_0) w_z = a w + s_1 x + s_0$

Mathematica

ClearAll["Global*"];
pde =  D[w[x,y,z],x]+(b1*x+b0)*D[w[x,y,z],y]+(c1*y+c0)*D[w[x,y,z],z]==a*w[x,y,z]+s1*x+s0;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to -\frac {a^2 \left (-e^{a x}\right ) c_1\left (-\text {b0} x-\frac {\text {b1} x^2}{2}+y,\frac {1}{2} \text {b0} \text {c1} x^2+\frac {1}{3} \text {b1} \text {c1} x^3-\text {c0} x-\text {c1} x y+z\right )+a \text {s0}+a \text {s1} x+\text {s1}}{a^2}\right \}\right \}$

Maple

restart;
local gamma;
pde :=  diff(w(x,y,z),x)+ (b__1*x+b__0)*diff(w(x,y,z),y)+ (c__1*x+c__0)*diff(w(x,y,z),z)=a*w(x,y,z)+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



$w \left ( x,y,z \right ) ={\frac {{{\rm e}^{ax}}{\it \_F1} \left ( -1/2\,b_{1}\,{x}^{2}-b_{0}\,x+y,-1/2\,c_{1}\,{x}^{2}-c_{0}\,x+z \right ) {a}^{2}+ \left ( -s_{1}\,x-s_{0} \right ) a-s_{1}}{{a}^{2}}}$

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##### 6.9.1.5 [1924] Problem 5

problem number 1924

Added Jan 6, 2020.

Problem Chapter 9.2.1.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + (a y+k_1 x+k_0) w_y + (b z+n_1 x+n_0) w_z = (c_1 x+c_0) w + s_1 x + s_0$

Mathematica

ClearAll["Global*"];
pde =  D[w[x,y,z],x]+(a*y+k1*x+k0)*D[w[x,y,z],y]+(b*z+n1*x+n0)*D[w[x,y,z],z]==(c1*x+c0)*w[x,y,z]+s1*x+s0;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (\frac {e^{-a x} \left (a^2 y+a (\text {k0}+\text {k1} x)+\text {k1}\right )}{a^2},\frac {e^{-b x} \left (b^2 z+b (\text {n0}+\text {n1} x)+\text {n1}\right )}{b^2}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {Erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}$

Maple

restart;
local gamma;
pde :=  diff(w(x,y,z),x)+ (a*y+k__1*x+k__0)*diff(w(x,y,z),y)+ (b*z+n__1*x+n__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



$w \left ( x,y,z \right ) =1/2\,{\frac {{{\rm e}^{1/2\,x \left ( c_{1}\,x+2\,c_{0} \right ) }}}{{c_{1}}^{5/2}} \left ( 2\,{\it \_F1} \left ( {\frac { \left ( y{a}^{2}+a \left ( k_{1}\,x+k_{0} \right ) +k_{1} \right ) {{\rm e}^{-ax}}}{{a}^{2}}},{\frac { \left ( z{b}^{2}+b \left ( n_{1}\,x+n_{0} \right ) +n_{1} \right ) {{\rm e}^{-bx}}}{{b}^{2}}} \right ) {c_{1}}^{5/2}+c_{1}\,\sqrt {2}\sqrt {\pi }{{\rm e}^{1/2\,{\frac {{c_{0}}^{2}}{c_{1}}}}} \left ( -c_{0}\,s_{1}+c_{1}\,s_{0} \right ) \erf \left ( 1/2\,\sqrt {2} \left ( x\sqrt {c_{1}}+{\frac {c_{0}}{\sqrt {c_{1}}}} \right ) \right ) -2\,{{\rm e}^{-1/2\,c_{1}\,{x}^{2}-c_{0}\,x}}{c_{1}}^{3/2}s_{1} \right ) }$

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##### 6.9.1.6 [1925] Problem 6

problem number 1925

Added Jan 6, 2020.

Problem Chapter 9.2.1.6, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + (a_2 y+a_1 x+a_0) w_y + (b_3 z+b_2 y+b_1 x+b_0) w_z = (c_3 z+c_2 y+c_1 x+c_0) w + s_3 z + s_2 y+s_1 x+s_0$

Mathematica

ClearAll["Global*"];
pde =  D[w[x,y,z],x]+(a2*y+a1*x+a0)*D[w[x,y,z],y]+(b3*z+b2*y+b1*x+b0)*D[w[x,y,z],z]==(c3*z+c2*y+c1*x+c0)*w[x,y,z]+s3*z+s2*y+s1*x+s0;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to \frac {u^{(0,1,0)}(x,y,z) (\text {a0}+\text {a1} x+\text {a2} y)+u^{(0,0,1)}(x,y,z) (\text {b0}+\text {b1} x+\text {b2} y+\text {b3} z)+u^{(1,0,0)}(x,y,z)-\text {s0}-\text {s1} x-\text {s2} y-\text {s3} z}{\text {c0}+\text {c1} x+\text {c2} y+\text {c3} z}\right \}\right \}$

Maple

restart;
local gamma;
pde :=  diff(w(x,y,z),x)+ (a__2*y+a__1*x+a__0)*diff(w(x,y,z),y)+ (b__3*z+b__2*y+b__1*x+b__0)*diff(w(x,y,z),z)=(c__3*z+c__2*y+c__1*x+c__0)*w(x,y,z)+s__3*z+s__2*y+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



time expired

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##### 6.9.1.7 [1926] Problem 7

problem number 1926

Added Jan 6, 2020.

Problem Chapter 9.2.1.7, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $a x w_x + b x w_y + c z w_z = (\alpha x+\beta ) w + p x+q$

Mathematica

ClearAll["Global*"];
pde =  a*x*D[w[x,y,z],x]+b*x*D[w[x,y,z],y]+c*z*D[w[x,y,z],z]==(alpha*x+beta)*w[x,y,z]+p*x+q;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (a \alpha x^{\frac {\beta }{a}} c_1\left (y-\frac {b x}{a},z x^{-\frac {c}{a}}\right )-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \text {Gamma}\left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \text {Gamma}\left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )\right )}{a \alpha }\right \}\right \}$

Maple

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



$w \left ( x,y,z \right ) =-3\,{\frac {1}{\alpha \, \left ( a-\beta \right ) \left ( 2\,a-\beta \right ) \left ( 3\,a-\beta \right ) \beta } \left ( \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}}q \left ( \alpha \,x+a-\beta \right ) {{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}{a}^{2} \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \left ( a-\beta /3 \right ) \WhittakerM \left ( 1/2\,{\frac {-a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -4/3\, \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}\beta \,{x}^{-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}} \left ( a-\beta /2 \right ) ^{2}ap \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {2\,a-\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) + \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}}q{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}a \left ( a-\beta \right ) ^{2} \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \left ( a-\beta /3 \right ) \WhittakerM \left ( 1/2\,{\frac {a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -2\, \left ( 1/3\, \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-{\frac {\beta }{a}}} \left ( 1/2\,\alpha \,x+a-\beta /2 \right ) {{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}}{a}^{2}p \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( -1/2\,{\frac {\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) + \left ( a-\beta /2 \right ) \alpha \, \left ( a-\beta \right ) {\it \_F1} \left ( {\frac {ay-bx}{a}},z{x}^{-{\frac {c}{a}}} \right ) \left ( a-\beta /3 \right ) \right ) \beta \right ) {x}^{{\frac {\beta }{a}}}{{\rm e}^{{\frac {\alpha \,x}{a}}}}}$

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##### 6.9.1.8 [1927] Problem 8

problem number 1927

Added Jan 6, 2020.

Problem Chapter 9.2.1.8, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $a x w_x + b y w_y + c z w_z = (\alpha x+\beta ) w + p x+q$

Mathematica

ClearAll["Global*"];
pde =  a*x*D[w[x,y,z],x]+b*y*D[w[x,y,z],y]+c*z*D[w[x,y,z],z]==(alpha*x+beta)*w[x,y,z]+p*x+q;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (a \alpha x^{\frac {\beta }{a}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \text {Gamma}\left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \text {Gamma}\left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )\right )}{a \alpha }\right \}\right \}$

Maple

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



$w \left ( x,y,z \right ) =-3\,{\frac {1}{\alpha \, \left ( a-\beta \right ) \left ( 2\,a-\beta \right ) \left ( 3\,a-\beta \right ) \beta }{x}^{{\frac {\beta }{a}}} \left ( \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}}q \left ( \alpha \,x+a-\beta \right ) {{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}{a}^{2} \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \left ( a-\beta /3 \right ) \WhittakerM \left ( 1/2\,{\frac {-a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -4/3\, \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}\beta \,{x}^{-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}} \left ( a-\beta /2 \right ) ^{2}ap \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {2\,a-\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) + \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}}q{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}a \left ( a-\beta \right ) ^{2} \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \left ( a-\beta /3 \right ) \WhittakerM \left ( 1/2\,{\frac {a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -2\,\beta \, \left ( 1/3\, \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-{\frac {\beta }{a}}} \left ( 1/2\,\alpha \,x+a-\beta /2 \right ) {{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}}{a}^{2}p \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( -1/2\,{\frac {\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) +{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) \left ( a-\beta /2 \right ) \alpha \, \left ( a-\beta \right ) \left ( a-\beta /3 \right ) \right ) \right ) {{\rm e}^{{\frac {\alpha \,x}{a}}}}}$

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##### 6.9.1.9 [1928] Problem 9

problem number 1928

Added Jan 6, 2020.

Problem Chapter 9.2.1.9, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $x w_x + a z w_y + b y w_z = (c x+k) w + p x+q$

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+k)*w[x,y,z]+p*x+q;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to \frac {e^{c x} \left (-(c x)^k (p \text {Gamma}(1-k,c x)+c q \text {Gamma}(-k,c x))+c x^k c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right )}{c}\right \}\right \}$

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+k)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));

`

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

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