6.7.4 2.4

6.7.4.1 [1603] Problem 1

problem number 1603

Added June 10, 2019.

Problem Chapter 7.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c w_z = \alpha x^n+ \beta y^m+ \gamma z^k \]

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

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

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

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

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6.7.4.2 [1604] Problem 2

problem number 1604

Added June 10, 2019.

Problem Chapter 7.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b y w_y + c z w_z = \alpha x^n+ \beta y^m+ \gamma z^k \]

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

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

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

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

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6.7.4.3 [1605] Problem 3

problem number 1605

Added June 10, 2019.

Problem Chapter 7.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a z w_y + b y w_z = c x^n \]

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

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

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

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

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6.7.4.4 [1606] Problem 4

problem number 1606

Added June 10, 2019.

Problem Chapter 7.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a x w_x + b y w_y + c z w_z = \alpha x^n + \beta y^m + \gamma z^k \]

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

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

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

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

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6.7.4.5 [1607] Problem 5

problem number 1607

Added June 10, 2019.

Problem Chapter 7.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ x w_x + a z w_y + b y w_z = c x^n \]

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

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

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

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

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6.7.4.6 [1608] Problem 6

problem number 1608

Added June 10, 2019.

Problem Chapter 7.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

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

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

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

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6.7.4.7 [1609] Problem 7

problem number 1609

Added June 10, 2019.

Problem Chapter 7.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a x^n y^m w_y + b x^\nu y^\mu z^\lambda w_z = c x^k \]

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

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

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6.7.4.8 [1610] Problem 8

problem number 1610

Added June 10, 2019.

Problem Chapter 7.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + (a_1 x^{n_1} y+ b_1 x^{m_1}) w_y + (a_2 x^{n_2} y+ b_2 x^{m_2}) w_z = c_2 x^{k_2} y + c_1 x^{k_1} z \]

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]== c2*x^k2*y+c1*x^k1*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 {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 )+\int _1^x\frac {(-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}} \left ((-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}} z K[2]^{\text {k1}}+(-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}} \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}} \text {c1} 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 {c1} (\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 {c1} \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 {c1} (\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 {c1} \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} \text {c1} 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} \text {c1} 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}} \text {c1} 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}} \text {c1} 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}} \text {c2} 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}} \text {c2} 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}} \text {c2} 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} \text {c2} 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} \text {c2} 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} \text {c2} 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} \text {c2} 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]\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)=c2*x^k2*y+c1*x^k1*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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6.7.4.9 [1611] Problem 9

problem number 1611

Added June 10, 2019.

Problem Chapter 7.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + (a_1 x^{n_1} y+ b_1 x^{m_1}) w_y + (a_2 x^{n_2} z+ b_2 x^{m_2}) w_z = c_2 x^{k_2} y + c_1 x^{k_1} z \]

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]== c2*x^k2*y+c1*x^k1*z; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \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 {c2} 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]+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 )\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)=c2*x^k2*y+c1*x^k1*z; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {1}{{\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}+1 \right ) {\it c2}\,{\it b1}\,{{\it \_a}}^{{\it k2}} \left ( {\it m2}+2\,{\it n2}+3 \right ) \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{\it a1}\,{x}^{{\it m1}+1} \right ) \left ( {\it m2}+{\it n2}+2 \right ) \left ( {\it n1}+1 \right ) ^{2}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{\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 ) {\it b2}\,{{\it \_a}}^{{\it k1}} \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 \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}}}}{\it c1}\, \left ( {\it m1}+1 \right ) {{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \left ( \left ( {\it m2}+{\it n2}+2 \right ) {x}^{-{\it n2}+{\it m2}}+{\it a2}\,{x}^{{\it m2}+1} \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) {\it a1}\, \left ( {\it n2}+1 \right ) ^{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}+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}}}{\it c2}\,{\it b1}\,{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\it m2}+2\,{\it n2}+3 \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 ) \left ( {\it m2}+{\it n2}+2 \right ) \left ( {\it n1}+1 \right ) ^{2}{\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 ( {{\rm e}^{{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{2\,{\it n2}+2}}}}{\it b2}\,{\it c1}\, \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 ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}}{\it a1}\, \left ( {\it n2}+1 \right ) ^{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 \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}} \right ) + \left ( -{\it c2}\,{{\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}\,{\it c1}\, \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}}}{\it a2}\,{{\it \_a}}^{{\it k2}-{\it n1}+{\it m1}}{\it c2}\,{\it b1}\,{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \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}\,{\it c1}\, \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 m2}+1 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) \left ( {\it c2}\,{{\it \_a}}^{{\it k2}}y{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}+{\it c1}\,{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}}}}z{{\it \_a}}^{{\it k1}} \right ) \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) {\it a1} \right ) \left ( {\it m2}+{\it n2}+2 \right ) \right ) \left ( {\it m1}+{\it n1}+2 \right ) \right ) }{d{\it \_a}}+{\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 ( -{\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 ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{\it a1}\,{x}^{{\it m1}+1} \right ) {{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\it n1}+1 \right ) ^{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 m1}+{\it n1}+2 \right ) \left ( -{{\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 b1}\,{x}^{-{\it n1}+{\it m1}} \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 ) \right ) },{\frac {1}{{\it a2}\, \left ( {\it m2}+1 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) \left ( {\it m2}+{\it n2}+2 \right ) } \left ( -{\it b2}\, \left ( {\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}} \left ( \left ( {\it m2}+{\it n2}+2 \right ) {x}^{-{\it n2}+{\it m2}}+{\it a2}\,{x}^{{\it m2}+1} \right ) {{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \left ( {\it n2}+1 \right ) ^{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 ( -{{\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 b2}\,{x}^{-{\it n2}+{\it m2}} \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 ) \right ) } \right ) \]

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6.7.4.10 [1612] Problem 10

problem number 1612

Added June 10, 2019.

Problem Chapter 7.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

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

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

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

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; 
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}{s+1} \left ( \left ( s+1 \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 ) +{{\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 ( \left ( -{\it n1}-2 \right ) {x}^{-{\it n1}}+{\it a1}\,x \left ( k-1 \right ) \right ) \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}\,{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \left ( {\it n1}+1 \right ) ^{2} \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 {{\it a1}\,{x}^{{\it n1}+1}k}{{\it n1}+1}}}} \left ( {\it n1}+2 \right ) {y}^{{\frac {{\it n1}}{{\it n1}+1}}} \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 {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{2\,{\it n2}+2}}}} \left ( -{\frac {{\it a2}\,{x}^{{\it n2}+1} \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 {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{{\it n2}+1}} \right ) +{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{2\,{\it n2}+2}}}}{z}^{{\frac {m}{{\it n2}+1}}}{z}^{{\frac {m{\it n2}}{{\it n2}+1}}} \left ( \left ( -{\it n2}-2 \right ) {x}^{-{\it n2}}+{\it a2}\,x \left ( m-1 \right ) \right ) \left ( -{\frac {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{{\it n2}+1}} \right ) ^{{\frac {-{\it n2}-2}{2\,{\it n2}+2}}}{\it b2}\,{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}} \left ( {\it n2}+1 \right ) ^{2} \WhittakerM \left ( -{\frac {{\it n2}}{2\,{\it n2}+2}},{\frac {2\,{\it n2}+3}{2\,{\it n2}+2}},-{\frac {{\it a2}\,{x}^{{\it n2}+1} \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 {{\it a2}\,{x}^{{\it n2}+1}m}{{\it n2}+1}}}} \left ( {\it n2}+2 \right ) {z}^{{\frac {{\it n2}}{{\it n2}+1}}} \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 ) +{x}^{s+1}c \right ) }\]

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6.7.4.11 [1613] Problem 11

problem number 1613

Added June 10, 2019.

Problem Chapter 7.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

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

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]== c1*x^s1+c2*y^s2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\left (\text {c1} K[1]^{\text {s1}}+\text {c2} \left (\left (\frac {(-1)^{-\frac {1}{\text {n1}+1}} \text {a1}^{-\frac {1}{\text {n1}+1}} \exp \left (-\frac {\text {a1} \left (x^{\text {n1}+1}+(k-1) K[1]^{\text {n1}+1}\right )}{\text {n1}+1}\right ) (k-1)^{-\frac {1}{\text {n1}+1}} y^{-k} \left (-\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (k-1) (\text {n1}+1)^{\frac {1}{\text {n1}+1}} \text {Gamma}\left (\frac {1}{\text {n1}+1},-\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}\right ) y^k+\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (k-1) (\text {n1}+1)^{\frac {1}{\text {n1}+1}} \text {Gamma}\left (\frac {1}{\text {n1}+1},-\frac {\text {a1} (k-1) K[1]^{\text {n1}+1}}{\text {n1}+1}\right ) y^k+(-1)^{\frac {1}{\text {n1}+1}} \text {a1}^{\frac {1}{\text {n1}+1}} e^{\frac {\text {a1} k x^{\text {n1}+1}}{\text {n1}+1}} (k-1)^{\frac {1}{\text {n1}+1}} (\text {n1}+1) y\right )}{\text {n1}+1}\right )^{\frac {1}{1-k}}\right )^{\text {s2}}\right )dK[1]+c_1\left (\text {b1} (-1)^{\frac {\text {n1}}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {1}{\text {n1}+1}} (k-1)^{\frac {\text {n1}}{\text {n1}+1}} \text {Gamma}\left (\frac {1}{\text {n1}+1},-\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}\right )+y^{1-k} e^{\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (-1)^{\frac {\text {n2}}{\text {n2}+1}} (\text {n2}+1)^{-\frac {\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {1}{\text {n2}+1}} (m-1)^{\frac {\text {n2}}{\text {n2}+1}} \text {Gamma}\left (\frac {1}{\text {n2}+1},-\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}\right )+z^{1-m} e^{\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]

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)= c1*x^s1+c2*y^s2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int ^{x}\!{\it c1}\,{{\it \_a}}^{{\it s1}}+{\it c2}\, \left ( \left ( {\frac {1}{{\it a1}\, \left ( {\it n1}+2 \right ) \left ( 2\,{\it n1}+3 \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 ) +{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 {{{\it \_a}}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{2\,{\it n1}+2}}}} \left ( -{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{{\it n1}+1}} \right ) ^{{\frac {-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\,{{\it \_a}}^{-{\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 {{{\it \_a}}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{{\it n1}+1}} \right ) +{{\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 ( \left ( -{\it n1}-2 \right ) {x}^{-{\it n1}}+{\it a1}\,x \left ( k-1 \right ) \right ) \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}\,{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \left ( {\it n1}+1 \right ) ^{2} \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 ) -{y}^{{\frac {k{\it n1}}{{\it n1}+1}}} \left ( \left ( -{\it n1}-2 \right ) {{\it \_a}}^{-{\it n1}}+{\it \_a}\,{\it a1}\, \left ( k-1 \right ) \right ) \left ( -{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{{\it n1}+1}} \right ) ^{{\frac {-{\it n1}-2}{2\,{\it n1}+2}}} \left ( {\it n1}+1 \right ) ^{2}{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}\, \left ( k-1 \right ) }{2\,{\it n1}+2}}}}{\it b1}\,{y}^{{\frac {k}{{\it n1}+1}}} \WhittakerM \left ( -{\frac {{\it n1}}{2\,{\it n1}+2}},{\frac {2\,{\it n1}+3}{2\,{\it n1}+2}},-{\frac {{{\it \_a}}^{{\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 {{\it a1}\,{x}^{{\it n1}+1}k}{{\it n1}+1}}}} \left ( {\it n1}+2 \right ) {y}^{{\frac {{\it n1}}{{\it n1}+1}}} \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}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \right ) ^{{\it s2}}{d{\it \_a}}+{\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 ) +{{\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 ( \left ( -{\it n1}-2 \right ) {x}^{-{\it n1}}+{\it a1}\,x \left ( k-1 \right ) \right ) \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}\,{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \left ( {\it n1}+1 \right ) ^{2} \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 {{\it a1}\,{x}^{{\it n1}+1}k}{{\it n1}+1}}}} \left ( {\it n1}+2 \right ) {y}^{{\frac {{\it n1}}{{\it n1}+1}}} \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 {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{2\,{\it n2}+2}}}} \left ( -{\frac {{\it a2}\,{x}^{{\it n2}+1} \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 {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{{\it n2}+1}} \right ) +{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{2\,{\it n2}+2}}}}{z}^{{\frac {m}{{\it n2}+1}}}{z}^{{\frac {m{\it n2}}{{\it n2}+1}}} \left ( \left ( -{\it n2}-2 \right ) {x}^{-{\it n2}}+{\it a2}\,x \left ( m-1 \right ) \right ) \left ( -{\frac {{\it a2}\,{x}^{{\it n2}+1} \left ( m-1 \right ) }{{\it n2}+1}} \right ) ^{{\frac {-{\it n2}-2}{2\,{\it n2}+2}}}{\it b2}\,{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}} \left ( {\it n2}+1 \right ) ^{2} \WhittakerM \left ( -{\frac {{\it n2}}{2\,{\it n2}+2}},{\frac {2\,{\it n2}+3}{2\,{\it n2}+2}},-{\frac {{\it a2}\,{x}^{{\it n2}+1} \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 {{\it a2}\,{x}^{{\it n2}+1}m}{{\it n2}+1}}}} \left ( {\it n2}+2 \right ) {z}^{{\frac {{\it n2}}{{\it n2}+1}}} \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 ) \]

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6.7.4.12 [1614] Problem 12

problem number 1614

Added June 10, 2019.

Problem Chapter 7.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ x w_x + y w_y + a \sqrt {x^2+y^2} w_z = b x^n \]

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

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

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

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

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6.7.4.13 [1615] Problem 13

problem number 1615

Added June 10, 2019.

Problem Chapter 7.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

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

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

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

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

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

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