6.6.4 2.4

6.6.4.1 [1430] Problem 1

problem number 1430

Added May 18, 2019.

Problem Chapter 6.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+ a x^n y^m w_y + b x^\nu y^\mu z^\lambda w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a x^{n+1}}{n+1}-(m-1)^{\frac {1}{m-1}} y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m,-\frac {b (m-1)^{\frac {\mu }{1-m}} x^{\nu +1} \left ((m-1)^{\frac {1}{m-1}} y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m\right )^{\frac {\mu }{1-m}} \left (\frac {(m-1)^{\frac {1}{m-1}} (n+1) y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m}{a x^{n+1}+(m-1)^{\frac {1}{m-1}} (n+1) y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m}\right )^{\frac {\mu }{m-1}} \operatorname {Hypergeometric2F1}\left (\frac {\mu }{m-1},\frac {\nu +1}{n+1},\frac {n+\nu +2}{n+1},\frac {a x^{n+1}}{(m-1)^{\frac {1}{m-1}} (n+1) y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m+a x^{n+1}}\right )}{\nu +1}-(\lambda -1)^{\frac {1}{\lambda -1}} z \left (\frac {(\lambda -1)^{\frac {1}{1-\lambda }}}{z}\right )^{\lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.4.2 [1431] Problem 2

problem number 1431

Added May 18, 2019.

Problem Chapter 6.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1}) w_y +(a_2 x^{n_2} y + b_2 x^{m_1}) w_z = 0 \]

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^m1)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
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^m1)*diff(w(x,y,z),z)= 0; 
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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6.6.4.3 [1432] Problem 3

problem number 1432

Added May 18, 2019.

Problem Chapter 6.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1}) w_y +(a_2 x^{n_2} z + b_2 x^{m_1}) w_z = 0 \]

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^m1)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (\text {n2}+1)^{\frac {\text {m1}-\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {\text {m1}+1}{\text {n2}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n2}+1},\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}\right )+z e^{-\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]

Maple ✓

restart; 
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^m1)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\frac {\left ({\mathrm e}^{-\frac {x \,x^{\operatorname {n1}} \operatorname {a1}}{2 \operatorname {n1} +2}} \operatorname {b1} \,x^{\operatorname {m1}} \left (\operatorname {n1} +1\right )^{2} \left (x \,x^{\operatorname {n1}} \operatorname {a1} +\operatorname {m1} +\operatorname {n1} +2\right ) \operatorname {WhittakerM}\left (\frac {-\operatorname {n1} +\operatorname {m1}}{2 \operatorname {n1} +2}, \frac {\operatorname {m1} +2 \operatorname {n1} +3}{2 \operatorname {n1} +2}, \frac {x \,x^{\operatorname {n1}} \operatorname {a1}}{\operatorname {n1} +1}\right )-\left (-{\mathrm e}^{-\frac {x \,x^{\operatorname {n1}} \operatorname {a1}}{2 \operatorname {n1} +2}} \operatorname {b1} \,x^{\operatorname {m1}} \left (\operatorname {n1} +1\right ) \left (\operatorname {m1} +\operatorname {n1} +2\right ) \operatorname {WhittakerM}\left (\frac {\operatorname {m1} +\operatorname {n1} +2}{2 \operatorname {n1} +2}, \frac {\operatorname {m1} +2 \operatorname {n1} +3}{2 \operatorname {n1} +2}, \frac {x \,x^{\operatorname {n1}} \operatorname {a1}}{\operatorname {n1} +1}\right )+{\mathrm e}^{-\frac {x \,x^{\operatorname {n1}} \operatorname {a1}}{\operatorname {n1} +1}} \left (\frac {x \,x^{\operatorname {n1}} \operatorname {a1}}{\operatorname {n1} +1}\right )^{\frac {\operatorname {m1} +\operatorname {n1} +2}{2 \operatorname {n1} +2}} y \,x^{\operatorname {n1}} \operatorname {a1} \left (\operatorname {m1} +1\right ) \left (\operatorname {m1} +2 \operatorname {n1} +3\right )\right ) \left (\operatorname {m1} +\operatorname {n1} +2\right )\right ) x^{-\operatorname {n1}} \left (\frac {x \,x^{\operatorname {n1}} \operatorname {a1}}{\operatorname {n1} +1}\right )^{\frac {-\operatorname {m1} -\operatorname {n1} -2}{2 \operatorname {n1} +2}}}{\operatorname {a1} \left (\operatorname {m1} +1\right ) \left (\operatorname {m1} +2 \operatorname {n1} +3\right ) \left (\operatorname {m1} +\operatorname {n1} +2\right )}, -\frac {\left ({\mathrm e}^{-\frac {x \,x^{\operatorname {n2}} \operatorname {a2}}{2 \operatorname {n2} +2}} \operatorname {b2} \,x^{\operatorname {m1}} \left (\operatorname {n2} +1\right )^{2} \left (x \,x^{\operatorname {n2}} \operatorname {a2} +\operatorname {m1} +\operatorname {n2} +2\right ) \operatorname {WhittakerM}\left (\frac {-\operatorname {n2} +\operatorname {m1}}{2 \operatorname {n2} +2}, \frac {\operatorname {m1} +2 \operatorname {n2} +3}{2 \operatorname {n2} +2}, \frac {x \,x^{\operatorname {n2}} \operatorname {a2}}{\operatorname {n2} +1}\right )-\left (-{\mathrm e}^{-\frac {x \,x^{\operatorname {n2}} \operatorname {a2}}{2 \operatorname {n2} +2}} \operatorname {b2} \,x^{\operatorname {m1}} \left (\operatorname {n2} +1\right ) \left (\operatorname {m1} +\operatorname {n2} +2\right ) \operatorname {WhittakerM}\left (\frac {\operatorname {m1} +\operatorname {n2} +2}{2 \operatorname {n2} +2}, \frac {\operatorname {m1} +2 \operatorname {n2} +3}{2 \operatorname {n2} +2}, \frac {x \,x^{\operatorname {n2}} \operatorname {a2}}{\operatorname {n2} +1}\right )+{\mathrm e}^{-\frac {x \,x^{\operatorname {n2}} \operatorname {a2}}{\operatorname {n2} +1}} \left (\frac {x \,x^{\operatorname {n2}} \operatorname {a2}}{\operatorname {n2} +1}\right )^{\frac {\operatorname {m1} +\operatorname {n2} +2}{2 \operatorname {n2} +2}} z \,x^{\operatorname {n2}} \operatorname {a2} \left (\operatorname {m1} +1\right ) \left (\operatorname {m1} +2 \operatorname {n2} +3\right )\right ) \left (\operatorname {m1} +\operatorname {n2} +2\right )\right ) x^{-\operatorname {n2}} \left (\frac {x \,x^{\operatorname {n2}} \operatorname {a2}}{\operatorname {n2} +1}\right )^{\frac {-\operatorname {m1} -\operatorname {n2} -2}{2 \operatorname {n2} +2}}}{\operatorname {a2} \left (\operatorname {m1} +1\right ) \left (\operatorname {m1} +2 \operatorname {n2} +3\right ) \left (\operatorname {m1} +\operatorname {n2} +2\right )}\right )\]

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6.6.4.4 [1433] Problem 4

problem number 1433

Added May 18, 2019.

Problem Chapter 6.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1}) w_y +(a_2 x^{n_2} z + b_2 y^{m_1}) w_z = 0 \]

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*y^m1)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
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*y^m1)*diff(w(x,y,z),z)= 0; 
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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6.6.4.5 [1434] Problem 5

problem number 1434

Added May 18, 2019.

Problem Chapter 6.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1} y^k1) w_y +(a_2 x^{n_2} z + b_2 x^{m_2} z^{k_2}) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = D[w[x, y,z], x] +(a1*x^n1*y+b1*x^m1*y^k1)*D[w[x, y,z], y] +(a2*x^n2*z+b2*x^m2*z^k1)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1*y^k1)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*x^m2*z^k1)*diff(w(x,y,z),z)= 0; 
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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6.6.4.6 [1435] Problem 6

problem number 1435

Added May 18, 2019.

Problem Chapter 6.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x+ b y^m w_y +c z^l w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = a*x^n*D[w[x, y,z], x] +b*y^m*D[w[x, y,z], y] +c*z^L*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*x^n*diff(w(x,y,z),x)+b*y^m*diff(w(x,y,z),y)+c*z^L*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \,x^{-n +1} \left (m -1\right )+a \,y^{1-m} \left (n -1\right )}{\left (n -1\right ) a}, \frac {-c \,x^{-n +1} \left (L -1\right )+a \,z^{1-L} \left (n -1\right )}{\left (n -1\right ) a}\right )\]

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6.6.4.7 [1436] Problem 7

problem number 1436

Added May 18, 2019.

Problem Chapter 6.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^m w_x+ b x^n w_y +c z^l w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*y^m*D[w[x, y,z], x] +b*x^n*D[w[x, y,z], y] +c*z^L*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  a*y^m*diff(w(x,y,z),x)+b*x^n*diff(w(x,y,z),y)+c*z^L*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \,x^{n +1} \left (1+m \right )+y^{1+m} \left (n +1\right ) a}{\left (n +1\right ) a}, \frac {c x \left (L -1\right ) {\left (\left (\frac {\textit {\_a}^{n +1} b \left (1+m \right )-b \,x^{n +1} \left (1+m \right )+y^{1+m} \left (n +1\right ) a}{\left (n +1\right ) a}\right )^{\frac {1}{1+m}}\right )}^{-m}+a \,z^{1-L}}{a}\right )\]

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6.6.4.8 [1437] Problem 8

problem number 1437

Added May 18, 2019.

Problem Chapter 6.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x(y^n - z^n) w_x+ y(z^n-x^n) w_y +z(x^n-y^n) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*(y^n-z^n)*D[w[x, y,z], x] +y*(z^n-x^n)*D[w[x, y,z], y] +z*(x^n-y^n)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  x*(y^n-z^n)*diff(w(x,y,z),x)+y*(z^n-x^n)*diff(w(x,y,z),y)+z*(x^n-y^n)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left (x , y , z\right ) = c_{4} c_{5} x^{\frac {c_{1}}{n}} \left (y^{n}\right )^{\frac {c_{1}}{n^{2}}} \left (z^{n}\right )^{\frac {c_{1}}{n^{2}}} {\mathrm e}^{\frac {-z^{n} c_{3} -y^{n} c_{3} -x^{n} c_{3} -c_{2} n +c_{1}}{n^{2}}}\]

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