6.8.4 2.4

6.8.4.1 [1774] Problem 1

problem number 1774

Added July 1, 2019.

Problem Chapter 8.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]== (lambda*x^n+beta*y^m+gamma*z^k)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {za-cx}{a}} \right ) {{\rm e}^{1/8\,{\frac {8\,a\beta \,c \left ( n+1 \right ) \left ( k+1 \right ) {y}^{m+1}+b \left ( m+1 \right ) \left ( a \left ( n+1 \right ) {z}^{k+1}+8\,\lambda \,{x}^{n+1}c \left ( k+1 \right ) \right ) }{ac \left ( k+1 \right ) b \left ( m+1 \right ) \left ( n+1 \right ) }}}}\]

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6.8.4.2 [1775] Problem 2

problem number 1775

Added July 1, 2019.

Problem Chapter 8.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]== (lambda*x^n+beta*y^m+gamma*z^k)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( y{{\rm e}^{-{\frac {bx}{a}}}},z{{\rm e}^{-{\frac {cx}{a}}}} \right ) {{\rm e}^{1/8\,{\frac {8\,{x}^{n+1}bck\lambda \,m+8\, \left ( n+1 \right ) \left ( \beta \,{y}^{m}ck+1/8\,{z}^{k}bm \right ) a}{ackbm \left ( n+1 \right ) }}}}\]

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6.8.4.3 [1776] Problem 3

problem number 1776

Added July 1, 2019.

Problem Chapter 8.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

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

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

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6.8.4.4 [1777] Problem 4

problem number 1777

Added July 1, 2019.

Problem Chapter 8.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]== (lambda*x^n+beta*y^m+gamma*z^k)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) {{\rm e}^{1/8\,\int ^{x}\!{\frac {1}{{\it \_a}\,a} \left ( 8\,\lambda \,{{\it \_a}}^{n}+8\,\beta \, \left ( y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) ^{m}+ \left ( z{x}^{-{\frac {c}{a}}}{{\it \_a}}^{{\frac {c}{a}}} \right ) ^{k} \right ) }{d{\it \_a}}}}\]

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6.8.4.5 [1778] Problem 5

problem number 1778

Added July 1, 2019.

Problem Chapter 8.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

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

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

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6.8.4.6 [1779] Problem 6

problem number 1779

Added July 1, 2019.

Problem Chapter 8.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}},x \left ( \left ( {\frac {y{a}^{2}\sqrt {2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+ \left ( {\frac {ya}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}+{\frac {bz}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) \sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}} \right ) ^{-1/2\,{\frac {a\sqrt {2}}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}} \right ) {{\rm e}^{-\int ^{y}\!-{\frac {c}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}b} \left ( x \left ( \left ( {\frac {y{a}^{2}\sqrt {2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+ \left ( {\frac {ya}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}+{\frac {bz}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) \sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}} \right ) ^{-1/2\,{\frac {a\sqrt {2}}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}} \left ( \left ( {\frac {{a}^{2}{\it \_a}\,\sqrt {2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+\sqrt {2\,{\frac {{{\it \_a}}^{2}{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+1}\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}} \right ) ^{1/2\,{\frac {a\sqrt {2}}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}} \right ) ^{n}{\frac {1}{\sqrt {2\,{\frac {{{\it \_a}}^{2}{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+1}}}}{d{\it \_a}}}}\]

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6.8.4.7 [1780] Problem 7

problem number 1780

Added July 1, 2019.

Problem Chapter 8.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left ( x,y,z \right ) ={{\rm e}^{{\frac {{x}^{k+1}c}{k+1}}}}{\it \_F1} \left ( {\frac { \left ( n+1 \right ) {y}^{-m+1}+{x}^{n+1}a \left ( m-1 \right ) }{n+1}},b \left ( \lambda -1 \right ) \int ^{x}\!{{\it \_a}}^{\nu } \left ( \left ( {\frac { \left ( n+1 \right ) {y}^{-m+1}+a \left ( {x}^{n+1}-{{\it \_a}}^{n+1} \right ) \left ( m-1 \right ) }{n+1}} \right ) ^{- \left ( m-1 \right ) ^{-1}} \right ) ^{\mu }{d{\it \_a}}+{z}^{1-\lambda } \right ) \]

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6.8.4.8 [1781] Problem 8

problem number 1781

Added July 1, 2019.

Problem Chapter 8.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^{n_1} y +b_1 x^{m_1} ) w_y + (a_2 x^{n_2} y +b_2 x^{m_2} ) w_z = (c_1 x^{k_2} y +c_1 x^{k_1} z ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x^n1*y+b1*x^m1)*D[w[x, y,z], y] +(a2*x^n2*y+b2*x^m2)*D[w[x,y,z],z]== (c1*x^k2*y+c1*x^k1*z)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\int _1^x\frac {(-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} \text {c1} e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \left ((-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} z K[2]^{\text {k1}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \text {n1}^2 z K[2]^{\text {k1}}+2 (-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \text {n1} z K[2]^{\text {k1}}+\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}+\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} \text {n1} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}-\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}-\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} \text {n1} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}-\text {a2} \text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}+\text {a2} \text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}+(-1)^{\frac {\text {n1}+\text {n2}+2}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^2 \int _1^x\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\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 ) K[1]^{\text {m1}}+\text {b2} K[1]^{\text {m2}}\right )dK[1] K[2]^{\text {k1}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^2 \int _1^{K[2]}\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\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 ) K[1]^{\text {m1}}+\text {b2} K[1]^{\text {m2}}\right )dK[1] K[2]^{\text {k1}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}} y K[2]^{\text {k2}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}} \text {n1}^2 y K[2]^{\text {k2}}+2 (-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}} \text {n1} y K[2]^{\text {k2}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} \text {n1} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}+(-1)^{\frac {\text {n1}+\text {n2}+2}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}+(-1)^{\frac {\text {n1}+\text {n2}+2}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} \text {n1} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}\right )}{(\text {n1}+1)^2}dK[2]} c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}}+e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} y,(-1)^{-\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}-\text {n1}}{\text {n1}+1}} y \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}}+z-\int _1^x\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\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 ) K[1]^{\text {m1}}+\text {b2} K[1]^{\text {m2}}\right )dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1)*diff(w(x,y,z),y)+(a2*x^n2*y+b2*x^m2)*diff(w(x,y,z),z)= (c1*x^k2*y+c1*x^k1*z)*w(x,y,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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6.8.4.9 [1782] Problem 9

problem number 1782

Added July 1, 2019.

Problem Chapter 8.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^{n_1} y +b_1 x^{m_1} ) w_y + (a_2 x^{n_2} z +b_2 x^{m_2} ) w_z = (c_1 x^{k_2} y +c_1 x^{k_1} z) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x^n1*y+b1*x^m1)*D[w[x, y,z], y] +(a2*x^n2*z+b2*x^m2)*D[w[x,y,z],z]== (c1*x^k2*y+c1*x^k1*z)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (\text {n2}+1)^{\frac {\text {m2}-\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {\text {m2}+1}{\text {n2}+1}} \text {Gamma}\left (\frac {\text {m2}+1}{\text {n2}+1},\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}\right )+z e^{-\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}}\right ) \exp \left (\int _1^x\left (\frac {\text {a2}^{-\frac {\text {m2}+1}{\text {n2}+1}} \text {c1} e^{\frac {\text {a2} \left (K[1]^{\text {n2}+1}-x^{\text {n2}+1}\right )}{\text {n2}+1}} \left ((\text {n2}+1) z \text {a2}^{\frac {\text {m2}+1}{\text {n2}+1}}+\text {b2} e^{\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}} (\text {n2}+1)^{\frac {\text {m2}+1}{\text {n2}+1}} \text {Gamma}\left (\frac {\text {m2}+1}{\text {n2}+1},\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}\right )-\text {b2} e^{\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}} (\text {n2}+1)^{\frac {\text {m2}+1}{\text {n2}+1}} \text {Gamma}\left (\frac {\text {m2}+1}{\text {n2}+1},\frac {\text {a2} K[1]^{\text {n2}+1}}{\text {n2}+1}\right )\right ) K[1]^{\text {k1}}}{\text {n2}+1}+\frac {\text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \text {c1} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} \left ((\text {n1}+1) y \text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}}+\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )-\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) K[1]^{\text {k2}}}{\text {n1}+1}\right )dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*x^m2)*diff(w(x,y,z),z)= (c1*x^k2*y+c1*x^k1*z)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) } \left ( - \left ( {\it n1}+1 \right ) ^{2}{\it b1}\, \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{\it a1}\,{x}^{{\it m1}+1} \right ) \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) + \left ( -{\it b1}\,{x}^{-{\it n1}+{\it m1}}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}y{\it a1}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \right ) \left ( {\it m1}+{\it n1}+2 \right ) \right ) },{\frac {1}{{\it a2}\, \left ( {\it m2}+2\,{\it n2}+3 \right ) \left ( {\it m2}+{\it n2}+2 \right ) \left ( {\it m2}+1 \right ) } \left ( - \left ( {\it n2}+1 \right ) ^{2}{\it b2}\, \left ( \left ( {\it m2}+{\it n2}+2 \right ) {x}^{-{\it n2}+{\it m2}}+{\it a2}\,{x}^{{\it m2}+1} \right ) \left ( {\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}}{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \WhittakerM \left ( {\frac {-{\it n2}+{\it m2}}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) + \left ( -{\it b2}\,{x}^{-{\it n2}+{\it m2}}{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \left ( {\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}} \left ( {\it n2}+1 \right ) \left ( {\it m2}+{\it n2}+2 \right ) \WhittakerM \left ( {\frac {{\it m2}+{\it n2}+2}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) +{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}z{\it a2}\, \left ( {\it m2}+1 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) \right ) \left ( {\it m2}+{\it n2}+2 \right ) \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac {{\it c1}}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) {\it a2}\, \left ( {\it m2}+2\,{\it n2}+3 \right ) \left ( {\it m2}+{\it n2}+2 \right ) \left ( {\it m2}+1 \right ) } \left ( - \left ( {\it m2}+{\it n2}+2 \right ) {{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\it m2}+2\,{\it n2}+3 \right ) {{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{\it b1}\, \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( {\it n1}+1 \right ) ^{2} \left ( {\it m2}+1 \right ) {{\it \_a}}^{{\it k2}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{\it a1}\,{x}^{{\it m1}+1} \right ) {\it a2}\, \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) - \left ( {\it m1}+{\it n1}+2 \right ) \left ( \left ( {\it m2}+{\it n2}+2 \right ) {x}^{-{\it n2}+{\it m2}}+{\it a2}\,{x}^{{\it m2}+1} \right ) {\it b2}\, \left ( {\it m1}+1 \right ) {{\it \_a}}^{{\it k1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}}}} \left ( {\it n2}+1 \right ) ^{2} \left ( {\it m1}+2\,{\it n1}+3 \right ) {\it a1}\,{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \left ( {\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}} \WhittakerM \left ( {\frac {-{\it n2}+{\it m2}}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) + \left ( {\it m2}+{\it n2}+2 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) {{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\, \left ( {\it n1}+1 \right ) ^{2} \left ( {\it m2}+1 \right ) \left ( \left ( {\it m1}+{\it n1}+2 \right ) {{\it \_a}}^{{\it k2}-{\it n1}+{\it m1}}+{{\it \_a}}^{{\it k2}+1+{\it m1}}{\it a1} \right ) {\it a2}\, \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) + \left ( {\it m1}+{\it n1}+2 \right ) \left ( {\it b2}\, \left ( {\it m1}+1 \right ) \left ( \left ( {\it m2}+{\it n2}+2 \right ) {{\it \_a}}^{-{\it n2}+{\it m2}+{\it k1}}+{{\it \_a}}^{{\it m2}+1+{\it k1}}{\it a2} \right ) \left ( {\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}} \left ( {\it n2}+1 \right ) ^{2} \left ( {\it m1}+2\,{\it n1}+3 \right ) {{\rm e}^{{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{2\,{\it n2}+2}}}}{\it a1}\, \WhittakerM \left ( {\frac {-{\it n2}+{\it m2}}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}} \right ) + \left ( {\it m2}+{\it n2}+2 \right ) \left ( -{{\it \_a}}^{{\it k2}}{x}^{-{\it n1}+{\it m1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it a2}\,{\it b1}\, \left ( {\it m2}+1 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) -{{\it \_a}}^{{\it k1}}{x}^{-{\it n2}+{\it m2}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}}}}{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \left ( {\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}}{\it a1}\,{\it b2}\, \left ( {\it n2}+1 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m2}+{\it n2}+2 \right ) \WhittakerM \left ( {\frac {{\it m2}+{\it n2}+2}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) + \left ( {\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}}{{\it \_a}}^{{\it k2}-{\it n1}+{\it m1}}{\it a2}\,{\it b1}\, \left ( {\it m2}+1 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) + \left ( {\it m1}+1 \right ) \left ( {{\it \_a}}^{-{\it n2}+{\it m2}+{\it k1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{2\,{\it n2}+2}}}} \left ( {\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}}{\it b2}\, \left ( {\it n2}+1 \right ) \left ( {\it m2}+{\it n2}+2 \right ) \WhittakerM \left ( {\frac {{\it m2}+{\it n2}+2}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}} \right ) +{\it a2}\, \left ( {{\it \_a}}^{{\it k1}}z{{\rm e}^{{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}}}}{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}+{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}y{{\it \_a}}^{{\it k2}} \right ) \left ( {\it m2}+1 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) {\it a1} \right ) \right ) \right ) }{d{\it \_a}}}}\]

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6.8.4.10 [1783] Problem 10

problem number 1783

Added July 1, 2019.

Problem Chapter 8.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^{n_1} y +b_1 y^k ) w_y + (a_2 x^{n_2} z +b_2 z^m ) w_z = c x^s w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x^n1*y+b1*y^k)*D[w[x, y,z], y] +(a2*x^n2*z+b2*z^m)*D[w[x,y,z],z]== c*x^s*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left (\text {b1} (-1)^{\frac {\text {n1}}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {1}{\text {n1}+1}} (k-1)^{\frac {\text {n1}}{\text {n1}+1}} \text {Gamma}\left (\frac {1}{\text {n1}+1},-\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}\right )+y^{1-k} e^{\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (-1)^{\frac {\text {n2}}{\text {n2}+1}} (\text {n2}+1)^{-\frac {\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {1}{\text {n2}+1}} (m-1)^{\frac {\text {n2}}{\text {n2}+1}} \text {Gamma}\left (\frac {1}{\text {n2}+1},-\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}\right )+z^{1-m} e^{\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+(a1*x^n1*y+b1*y^k)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*z^m)*diff(w(x,y,z),z)= c*x^s*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{{\it a1}\, \left ( 2\,{{\it n1}}^{2}+7\,{\it n1}+6 \right ) } \left ( -{y}^{{\frac {k}{{\it n1}+1}}}{y}^{{\frac {k{\it n1}}{{\it n1}+1}}}{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{2\,{\it n1}+2}}}} \left ( -{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{{\it n1}+1}} \right ) ^{{\frac {-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\,{x}^{-{\it n1}} \left ( {\it n1}+1 \right ) \left ( {\it n1}+2 \right ) ^{2} \WhittakerM \left ( {\frac {{\it n1}+2}{2\,{\it n1}+2}},{\frac {2\,{\it n1}+3}{2\,{\it n1}+2}},-{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{{\it n1}+1}} \right ) + \left ( {\it n1}+1 \right ) ^{2}{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{2\,{\it n1}+2}}}}{y}^{{\frac {k}{{\it n1}+1}}}{y}^{{\frac {k{\it n1}}{{\it n1}+1}}} \left ( -{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{{\it n1}+1}} \right ) ^{{\frac {-{\it n1}-2}{2\,{\it n1}+2}}} \left ( \left ( -{\it n1}-2 \right ) {x}^{-{\it n1}}+x{\it a1}\, \left ( k-1 \right ) \right ) {\it b1}\,{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \WhittakerM \left ( -{\frac {{\it n1}}{2\,{\it n1}+2}},{\frac {2\,{\it n1}+3}{2\,{\it n1}+2}},-{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{{\it n1}+1}} \right ) +2\,{\it a1}\,{y}^{ \left ( {\it n1}+1 \right ) ^{-1}} \left ( {\it n1}+3/2 \right ) {{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}\,k}{{\it n1}+1}}}}{y}^{{\frac {{\it n1}}{{\it n1}+1}}} \left ( {\it n1}+2 \right ) \right ) \left ( {y}^{{\frac {k{\it n1}}{{\it n1}+1}}} \right ) ^{-1} \left ( {y}^{{\frac {k}{{\it n1}+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \right ) ^{-1}},{\frac {1}{{\it a2}\, \left ( 2\,{{\it n2}}^{2}+7\,{\it n2}+6 \right ) } \left ( -{z}^{{\frac {m}{{\it n2}+1}}}{z}^{{\frac {m{\it n2}}{{\it n2}+1}}}{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}{{\rm e}^{{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( m-1 \right ) }{2\,{\it n2}+2}}}} \left ( -{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( m-1 \right ) }{{\it n2}+1}} \right ) ^{{\frac {-{\it n2}-2}{2\,{\it n2}+2}}}{\it b2}\,{x}^{-{\it n2}} \left ( {\it n2}+1 \right ) \left ( {\it n2}+2 \right ) ^{2} \WhittakerM \left ( {\frac {{\it n2}+2}{2\,{\it n2}+2}},{\frac {2\,{\it n2}+3}{2\,{\it n2}+2}},-{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( m-1 \right ) }{{\it n2}+1}} \right ) + \left ( {\it n2}+1 \right ) ^{2}{{\rm e}^{{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( m-1 \right ) }{2\,{\it n2}+2}}}}{z}^{{\frac {m}{{\it n2}+1}}}{z}^{{\frac {m{\it n2}}{{\it n2}+1}}} \left ( -{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( m-1 \right ) }{{\it n2}+1}} \right ) ^{{\frac {-{\it n2}-2}{2\,{\it n2}+2}}} \left ( \left ( -{\it n2}-2 \right ) {x}^{-{\it n2}}+x{\it a2}\, \left ( m-1 \right ) \right ) {\it b2}\,{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}} \WhittakerM \left ( -{\frac {{\it n2}}{2\,{\it n2}+2}},{\frac {2\,{\it n2}+3}{2\,{\it n2}+2}},-{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( m-1 \right ) }{{\it n2}+1}} \right ) +2\,{\it a2}\,{z}^{ \left ( {\it n2}+1 \right ) ^{-1}} \left ( {\it n2}+3/2 \right ) {{\rm e}^{{\frac {{x}^{{\it n2}+1}{\it a2}\,m}{{\it n2}+1}}}}{z}^{{\frac {{\it n2}}{{\it n2}+1}}} \left ( {\it n2}+2 \right ) \right ) \left ( {z}^{{\frac {m{\it n2}}{{\it n2}+1}}} \right ) ^{-1} \left ( {z}^{{\frac {m}{{\it n2}+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}} \right ) ^{-1}} \right ) {{\rm e}^{{\frac {{x}^{s+1}c}{s+1}}}}\]

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6.8.4.11 [1784] Problem 11

problem number 1784

Added July 1, 2019.

Problem Chapter 8.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^{n_1} y +b_1 y^k ) w_y + (a_2 y^{n_2} z +b_2 z^m ) w_z = (c_1 x^{s_1} + c_2 y^{s_2} + c_3 z^{s_3}) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x^n1*y+b1*y^k)*D[w[x, y,z], y] +(a2*y^n2*z+b2*z^m)*D[w[x,y,z],z]== (c1*x^s1+c2*y^s2+c3*z^s3)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+(a1*x^n1*y+b1*y^k)*diff(w(x,y,z),y)+(a2*y^n2*z+b2*z^m)*diff(w(x,y,z),z)= (c1*x^s1+c2*y^s2+c3*z^s3)*w(x,y,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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6.8.4.12 [1785] Problem 12

problem number 1785

Added July 1, 2019.

Problem Chapter 8.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ x w_x + y w_y + a \sqrt {x^2+y^2} w_z = b x^n w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y,z], x] + y*D[w[x, y,z], y] +a*Sqrt[x^2+y^2]*D[w[x,y,z],z]== b*x^n*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde :=  x*diff(w(x,y,z),x)+y*diff(w(x,y,z),y)+a*sqrt(x^2+y^2)*diff(w(x,y,z),z)= b*x^n*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},-a\sqrt {{x}^{2}+{y}^{2}}+z \right ) {{\rm e}^{{\frac {b{x}^{n}}{n}}}}\]

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6.8.4.13 [1786] Problem 13

problem number 1786

Added July 1, 2019.

Problem Chapter 8.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ x w_x + y w_y + (z - a \sqrt {x^2+y^2+z^2} w_z = b x^n w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y,z], x] + y*D[w[x, y,z], y] +(z-a*Sqrt[x^2+y^2+z^2])*D[w[x,y,z],z]== b*x^n*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {b x^n}{n}} c_1\left (\frac {y}{x},\log \left (-\sqrt {\frac {x^{2 a} \left (y^2+2 z^2\right )+x^{2 a+2}-2 \sqrt {z^2 x^{4 a} \left (x^2+y^2+z^2\right )}}{x^2+y^2}}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  x*diff(w(x,y,z),x)+y*diff(w(x,y,z),y)+(z-a*sqrt(x^2+y^2+z^2))*diff(w(x,y,z),z)= b*x^n*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}}, \left ( z+\sqrt {{x}^{2}+{y}^{2}+{z}^{2}} \right ) {x}^{a-1} \right ) {{\rm e}^{{\frac {b{x}^{n}}{n}}}}\]

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