7.7.4 2.4

7.7.4.1 [1604] Problem 1

problem number 1604

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'));
 

\[w \left (x , y , z\right ) = \frac {\left (m +1\right ) \left (k +1\right ) \left (n +1\right ) a b c \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a z -c x}{a}\right )+\left (m +1\right ) \left (k +1\right ) \alpha b c \,x^{n +1}+\left (n +1\right ) \left (\left (m +1\right ) b \gamma \,z^{k +1}+\left (k +1\right ) \beta c \,y^{m +1}\right ) a}{\left (m +1\right ) \left (k +1\right ) \left (n +1\right ) a b c}\]

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7.7.4.2 [1605] Problem 2

problem number 1605

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'));
 

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

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7.7.4.3 [1606] Problem 3

problem number 1606

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]];
 

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

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

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

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7.7.4.4 [1607] Problem 4

problem number 1607

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'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\frac {\alpha \,\textit {\_a}^{n}+\beta \left (y \,\textit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )^{m}+\gamma \left (z \,\textit {\_a}^{\frac {c}{a}} x^{-\frac {c}{a}}\right )^{k}}{\textit {\_a} a}d \textit {\_a} +\textit {\_F1} \left (y \,x^{-\frac {b}{a}}, z \,x^{-\frac {c}{a}}\right )\]

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7.7.4.5 [1608] Problem 5

problem number 1608

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'));
 

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

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7.7.4.6 [1609] Problem 6

problem number 1609

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]];
 

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

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'));
 

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

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7.7.4.7 [1610] Problem 7

problem number 1610

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'));
 

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

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7.7.4.8 [1611] Problem 8

problem number 1611

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}} \operatorname {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 \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}}+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}} \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \operatorname {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}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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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7.7.4.9 [1612] Problem 9

problem number 1612

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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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'));
 

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

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7.7.4.10 [1613] Problem 10

problem number 1613

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]];
 

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

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

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

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7.7.4.11 [1614] Problem 11

problem number 1614

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]];
 

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

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}\left (\mathit {c1} \,\textit {\_a}^{\mathit {s1}}+\mathit {c2} \left (\left (\frac {\left (\mathit {n1} +1\right ) \left (\mathit {n1} +2\right )^{2} \mathit {b1} \,\textit {\_a}^{-\mathit {n1}} \left (-\frac {\left (k -1\right ) \mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{2 \mathit {n1} +2}}-\left (\mathit {n1} +1\right ) \left (\mathit {n1} +2\right )^{2} \mathit {b1} \,x^{-\mathit {n1}} \left (-\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\left (k -1\right ) \mathit {a1} x +\left (-\mathit {n1} -2\right ) x^{-\mathit {n1}}\right ) \left (\mathit {n1} +1\right )^{2} \mathit {b1} \left (-\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (-\frac {\mathit {n1}}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}-\left (\left (k -1\right ) \textit {\_a} \mathit {a1} +\left (-\mathit {n1} -2\right ) \textit {\_a}^{-\mathit {n1}}\right ) \left (\mathit {n1} +1\right )^{2} \mathit {b1} \left (-\frac {\left (k -1\right ) \mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (-\frac {\mathit {n1}}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+2 \left (\mathit {n1} +2\right ) \left (\mathit {n1} +\frac {3}{2}\right ) \mathit {a1} \,y^{-k +1} {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}}{\left (2 \mathit {n1} +3\right ) \left (\mathit {n1} +2\right ) \mathit {a1}}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}}\right )^{\mathit {s2}}\right )d \textit {\_a} +\textit {\_F1} \left (\frac {-\left (\mathit {n1} +1\right ) \left (\mathit {n1} +2\right )^{2} \mathit {b1} \,x^{-\mathit {n1}} \left (-\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\left (k -1\right ) \mathit {a1} x +\left (-\mathit {n1} -2\right ) x^{-\mathit {n1}}\right ) \left (\mathit {n1} +1\right )^{2} \mathit {b1} \left (-\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (-\frac {\mathit {n1}}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+2 \left (\mathit {n1} +2\right ) \left (\mathit {n1} +\frac {3}{2}\right ) \mathit {a1} \,y^{-k +1} {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}}{\left (2 \mathit {n1}^{2}+7 \mathit {n1} +6\right ) \mathit {a1}}, \frac {-\left (\mathit {n2} +1\right ) \left (\mathit {n2} +2\right )^{2} \mathit {b2} \,x^{-\mathit {n2}} \left (-\frac {\left (m -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (\frac {\mathit {n2} +2}{2 \mathit {n2} +2}, \frac {2 \mathit {n2} +3}{2 \mathit {n2} +2}, -\frac {\left (m -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{\frac {\left (m -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+\left (\left (m -1\right ) \mathit {a2} x +\left (-\mathit {n2} -2\right ) x^{-\mathit {n2}}\right ) \left (\mathit {n2} +1\right )^{2} \mathit {b2} \left (-\frac {\left (m -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (-\frac {\mathit {n2}}{2 \mathit {n2} +2}, \frac {2 \mathit {n2} +3}{2 \mathit {n2} +2}, -\frac {\left (m -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{\frac {\left (m -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+2 \left (\mathit {n2} +2\right ) \left (\mathit {n2} +\frac {3}{2}\right ) \mathit {a2} \,z^{-m +1} {\mathrm e}^{\frac {\left (m -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}}}{\left (2 \mathit {n2}^{2}+7 \mathit {n2} +6\right ) \mathit {a2}}\right )\]

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7.7.4.12 [1615] Problem 12

problem number 1615

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]];
 

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

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

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

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7.7.4.13 [1616] Problem 13

problem number 1616

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]];
 

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

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

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

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