6.3.5 2.4

6.3.5.1 [866] Problem 1

problem number 866

Added Feb. 9, 2019.

Problem Chapter 3.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.2 [867] Problem 2

problem number 867

Added Feb. 9, 2019.

Problem Chapter 3.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {-c x^{n+1} (b x-a (n+2) y)+a^2 \left (n^2+3 n+2\right ) c_1\left (y-\frac {b x}{a}\right )}{a^2 (n+1) (n+2)}\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.3 [868] Problem 3

problem number 868

Added Feb. 9, 2019.

Problem Chapter 3.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.4 [869] Problem 4

problem number 869

Added Feb. 9, 2019.

Problem Chapter 3.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.5 [870] Problem 5

problem number 870

Added Feb. 9, 2019.

Problem Chapter 3.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

Result has unresolved integral

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.6 [871] Problem 6

problem number 871

Added Feb. 9, 2019.

Problem Chapter 3.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

Maple ✓

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

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

Result has unresolved integral

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.7 [872] Problem 7

problem number 872

Added Feb. 9, 2019.

Problem Chapter 3.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

Maple ✓

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

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

Result has unresolved integral

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.8 [873] Problem 8

problem number 873

Added Feb. 9, 2019.

Problem Chapter 3.2.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

Maple ✓

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

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

Result has unresolved integral

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.9 [874] Problem 9

problem number 874

Added Feb. 9, 2019.

Problem Chapter 3.2.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.10 [875] Problem 10

problem number 875

Added Feb. 9, 2019.

Problem Chapter 3.2.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.3.5.11 [876] Problem 11

problem number 876

Added Feb. 9, 2019.

Problem Chapter 3.2.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b x^m w_y = c x^m + d \]

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________