6.4.5 2.4

6.4.5.1 [1044] Problem 1

problem number 1044

Added Feb. 17, 2019.

Problem Chapter 4.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (c x^n + d y^m) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.5.2 [1045] Problem 2 case \(n\neq -1, n\neq -2\)

problem number 1045

Added Feb. 17, 2019.

Problem Chapter 4.2.4.2 case \(n eq -1, n eq -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
sol[[2]] = Assuming[{n != -1, n != -2}, Simplify[sol[[2]]]];
 

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

Maple ✓

restart; 
pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) =  c*x^n*y*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol)  assuming n<>-1, n<>-2;
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) {\mathrm e}^{\frac {x^{n +1} c \left (y \left (n +2\right ) a -b x \right )}{a^{2} \left (n +1\right ) \left (n +2\right )}}\]

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6.4.5.3 [1046] Problem 2 case \(n = -1\)

problem number 1046

Added Feb. 17, 2019.

Problem Chapter 4.2.4.2 case \(n= -1\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> n == -1], 60*10]];
 

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

Maple ✓

restart; 
pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) =  c*x^n*y*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n=-1),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) x^{\frac {c \left (y a -b x \right )}{a^{2}}} {\mathrm e}^{\frac {c x b}{a^{2}}}\]

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6.4.5.4 [1047] Problem 2 case \(n = -2\)

problem number 1047

Added Feb. 17, 2019.

Problem Chapter 4.2.4.2 case \(n= -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) x^{\frac {c b}{a^{2}}} {\mathrm e}^{-\frac {c \left (y a -b x \right )}{a^{2} x}}\]

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6.4.5.5 [1048] Problem 3

problem number 1048

Added Feb. 17, 2019.

Problem Chapter 4.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a(x^2+y^2)^k w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {a \left (x^2+y^2\right )^k}{2 k}}\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y}{x}\right ) {\mathrm e}^{\frac {a \left (x^{2}+y^{2}\right )^{k}}{2 k}}\]

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6.4.5.6 [1049] Problem 4

problem number 1049

Added Feb. 17, 2019.

Problem Chapter 4.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c x^n y^m w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.5.7 [1050] Problem 5

problem number 1050

Added Feb. 17, 2019.

Problem Chapter 4.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = (c x^n + k y^m) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.5.8 [1051] Problem 6

problem number 1051

Added Feb. 17, 2019.

Problem Chapter 4.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ m x w_x + n y w_y = (a x^n + b y^m)^k w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {n}{m}}\right ) \exp \left (\int _1^x\frac {\left (b \left (x^{-\frac {n}{m}} y K[1]^{\frac {n}{m}}\right )^m+a K[1]^n\right )^k}{m K[1]}dK[1]\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.5.9 [1052] Problem 7

problem number 1052

Added Feb. 17, 2019.

Problem Chapter 4.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + b y^m w_y = (c x^k + d y^s) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-b \,x^{-n +1} \left (m -1\right )+y^{1-m} a \left (n -1\right )}{\left (n -1\right ) a}\right ) {\mathrm e}^{-\frac {\left (x^{n} y^{1-m} \left (x^{n}\right )^{\frac {s}{m -1}} \left (n -1\right )^{\frac {s}{m -1}} a^{\frac {m -1+s}{m -1}} \left (y^{1-m} x^{n} a \left (n -1\right )\right )^{-\frac {s}{m -1}} d \left (1+k -n \right ) {\mathrm e}^{\frac {i s \pi \left (-\operatorname {csgn}\left (i a \,y^{1-m} x^{n} \left (n -1\right )\right )-\operatorname {csgn}\left (\frac {i}{n -1}\right )+\operatorname {csgn}\left (i y^{1-m} x^{n} a \right ) \operatorname {csgn}\left (i a \,y^{1-m} x^{n} \left (n -1\right )\right ) \operatorname {csgn}\left (\frac {i}{n -1}\right )+\operatorname {csgn}\left (i y^{1-m} x^{n} a \right ) \operatorname {csgn}\left (i a \,y^{1-m}\right ) \operatorname {csgn}\left (i x^{-n}\right )+\operatorname {csgn}\left (i y^{1-m}\right )-\operatorname {csgn}\left (\frac {i}{a}\right )+\operatorname {csgn}\left (i a \,y^{1-m}\right ) \operatorname {csgn}\left (i y^{1-m}\right ) \operatorname {csgn}\left (\frac {i}{a}\right )-\operatorname {csgn}\left (i x^{-n}\right )\right )}{2 m -2}}-c \,x^{k +1} b \left (m -s -1\right )\right ) x^{-n}}{\left (1+k -n \right ) b \left (m -s -1\right ) a}}\]

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6.4.5.10 [1053] Problem 8

problem number 1053

Added Feb. 17, 2019.

Problem Chapter 4.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + b x^m y w_y = (c x^k y^s + d) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{a \left (m -n +1\right )}}\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (c \,\textit {\_a}^{k} \left (y \,{\mathrm e}^{\frac {b \,\textit {\_a}^{-n} \left (-x^{m -n +1} \textit {\_a}^{n}+\textit {\_a}^{1+m}\right )}{a \left (m -n +1\right )}}\right )^{s}+d \right ) \textit {\_a}^{-n}d \textit {\_a}}{a}}\]

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6.4.5.11 [1054] Problem 9

problem number 1054

Added Feb. 17, 2019.

Problem Chapter 4.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + (b x^m y+c x^k) w_y = (s x^p y^q + d) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^n*D[w[x, y], x] + (b*x^m*y + c*x^k)*D[w[x, y], y] == (s*x^p*y^q + d)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (c (a (m-n+1))^{\frac {k-m}{m-n+1}} b^{\frac {-k+n-1}{m-n+1}} \Gamma \left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\int _1^x\frac {K[1]^{-n} \left (s \left (b^{-\frac {k+1}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \Gamma \left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}-b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \Gamma \left (\frac {k-n+1}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}+b^{\frac {k+1}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^p+d\right )}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.5.12 [1055] Problem 10

problem number 1055

Added Feb. 17, 2019.

Problem Chapter 4.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + b x^m y^k w_y = (c x^p y^q + s) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\frac {y^{1-k} (-m+n-1)}{k-1}-\frac {b x^{m-n+1}}{a}}{m-n+1}\right ) \exp \left (\frac {x^{1-n} \left (-\frac {c x^p \left (\left (y^{k-1}\right )^{\frac {1}{k-1}}\right )^q \left (\frac {a y (m-n+1) x^n}{a y (m-n+1) x^n+b (k-1) y^k x^{m+1}}\right )^{\frac {q}{k-1}} \operatorname {Hypergeometric2F1}\left (\frac {-n+p+1}{m-n+1},\frac {q}{k-1},\frac {m-2 n+p+2}{m-n+1},\frac {b (k-1) x^{m+1} y^k}{b (k-1) x^{m+1} y^k+a (m-n+1) x^n y}\right )}{n-p-1}-\frac {s}{n-1}\right )}{a}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {x^{m -n +1} \left (k -1\right ) b +y^{1-k} \left (m -n +1\right ) a}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (c \,\textit {\_a}^{p} {\left (\left (\frac {\textit {\_a}^{-n} \left (b \left (k -1\right ) \textit {\_a}^{n} x^{m -n +1}+a \,\textit {\_a}^{n} \left (m -n +1\right ) y^{1-k}-b \,\textit {\_a}^{1+m} \left (k -1\right )\right )}{\left (m -n +1\right ) a}\right )^{-\frac {1}{k -1}}\right )}^{q}+s \right ) \textit {\_a}^{-n}d \textit {\_a}}{a}}\]

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6.4.5.13 [1056] Problem 11

problem number 1056

Added Feb. 17, 2019.

Problem Chapter 4.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b x^n w_y = (c x^m + s) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right ) \exp \left (\frac {x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \left (c x^m \operatorname {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {m+1}{n+1},\frac {m+n+2}{n+1},\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )+(m+1) s \operatorname {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},1+\frac {1}{n+1},\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )\right )}{a (m+1)}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-b \,x^{n +1} \left (1+k \right )+y^{1+k} \left (n +1\right ) a}{\left (n +1\right ) a}\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (c \,\textit {\_a}^{m}+s \right ) {\left (\left (\frac {b \left (1+k \right ) \textit {\_a}^{n +1}-b \,x^{n +1} \left (1+k \right )+y^{1+k} \left (n +1\right ) a}{\left (n +1\right ) a}\right )^{\frac {1}{1+k}}\right )}^{-k}d \textit {\_a}}{a}}\]

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6.4.5.14 [1057] Problem 12

problem number 1057

Added Feb. 17, 2019.

Problem Chapter 4.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x(x^n + (a n -1) y^n) w_x + y(y^n + (a n -1) x^n) w_y = k n (x^n + y^n) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*(x^n + (a*n - 1)*y^n)*D[w[x, y], x] + y*(y^n + (a*n - 1)*x^n)*D[w[x, y], y] == k*n*(x^n + y^n)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := x*(x^n + (a*n -1)*y^n)*diff(w(x,y),x)+y*(y^n + (a*n -1)*x^n)*diff(w(x,y),y) =   k*n*(x^n + y^n)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

time expired

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6.4.5.15 [1058] Problem 13

problem number 1058

Added Feb. 17, 2019.

Problem Chapter 4.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x((n-2) y^n -2 x^n ) w_x + y(2 y^n - (n-2) x^n) w_y =\left ( (a(n-2)+2 b) y^n - (2 a + b(n-2)) x^n) \right ) w \]

Mathematica ✗

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

Failed

Maple ✗

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

time expired

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