59 HFOPDE, chapter 2.7.3

____________________________________________________________________________________

59.1 problem number 1

problem number 536

Added January 29, 2019.

Problem 2.7.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arctan ^k(\lambda x)+b\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (a*ArcTan[lambda*x]^k + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+( a*arctan(lambda*x)^k+b  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx-\int \!a \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+y \right ) \]

____________________________________________________________________________________

59.2 problem number 2

problem number 537

Added January 29, 2019.

Problem 2.7.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arctan ^k(\lambda y)+b\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (a*ArcTan[lambda*y]^k + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+( a*arctan(lambda*y)^k+b  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \arctan \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]

____________________________________________________________________________________

59.3 problem number 3

problem number 538

Added January 29, 2019.

Problem 2.7.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + k \arctan ^n(a x+b y+c) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + k*ArcTan[a*x + b*y + c]^n*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+k *arctan(a*x+b*y+c)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {ax+by}{b}}}\! \left ( k \left ( \arctan \left ( b{\it \_a}+c \right ) \right ) ^{n}b+a \right ) ^{-1}{d{\it \_a}}b+x \right ) \]

____________________________________________________________________________________

59.4 problem number 4

problem number 539

Added January 29, 2019.

Problem 2.7.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \arctan ^k(\lambda x) \arctan ^n(\mu y) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + a*ArcTan[lambda*x]^k*ArcTan[mu*y]^n*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+a*arctan(lambda*x)^k*arctan(mu*y)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \arctan \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \]

____________________________________________________________________________________

59.5 problem number 5

problem number 540

Added January 29, 2019.

Problem 2.7.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (y^2 + \lambda (\arctan x)^n y -a^2 + a \lambda (\arctan x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (y^2 + lambda*ArcTan[x]^n*y - a^2 + a*lambda*ArcTan[x]^n)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+(y^2 + lambda*arctan(x)^n*y -a^2 + a *lambda*arctan(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {y\int \!{{\rm e}^{\lambda \,\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x-2\,ax}}\,{\rm d}x+\int \!{{\rm e}^{\lambda \,\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x-2\,ax}}\,{\rm d}xa+{{\rm e}^{\lambda \,\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x-2\,ax}}}{y+a}} \right ) \]

____________________________________________________________________________________

59.6 problem number 6

problem number 541

Added January 29, 2019.

Problem 2.7.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (y^2 + \lambda x (\arctan x)^n y + \lambda (\arctan x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (y^2 + lambda*x*ArcTan[x]^n*y + lambda*ArcTan[x]^n)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+(y^2 + lambda*x*arctan(x)^n*y + lambda*arctan(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]

____________________________________________________________________________________

59.7 problem number 7

problem number 542

Added Feb. 1, 2019.

Problem 2.7.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x - \left ( (k+1)x^k y^2 - \lambda (\arctan x)^n (x^{k+1}y-1)\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcTan[x]^n*(x^(k + 1)*y - 1))*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)-((k+1)*x^k*y^2 - lambda*arctan(x)^n*(x^(k+1)*y-1)  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{x}^{k+1}y-1} \left ( y{x}^{k+1}\int \!{\frac {{{\rm e}^{\lambda \,\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk+y{x}^{k+1}\int \!{\frac {{{\rm e}^{\lambda \,\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x-\int \!{\frac {{{\rm e}^{\lambda \,\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk-{x}^{k+1}{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{k+1}\lambda \,x-2\,k-2}{x}}\,{\rm d}x}}-\int \!{\frac {{{\rm e}^{\lambda \,\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \right ) } \right ) \]

____________________________________________________________________________________

59.8 problem number 8

problem number 543

Added Feb. 1, 2019.

Problem 2.7.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arctan x)^n +a y+ a b - b^2 \lambda (\arctan x)^n n\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (lambda*ArcTan[x]^n + a*y + a*b - b^2*lambda*ArcTan[x]^n*n)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+(lambda* arctan(x)^n +a*y+ a*b - b^2*lambda*arctan(x)^n*n  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \int \!{{\rm e}^{-ax}} \left ( {b}^{2}\lambda \, \left ( \arctan \left ( x \right ) \right ) ^{n}n- \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda -ab \right ) \,{\rm d}x+y{{\rm e}^{-ax}} \right ) \]

____________________________________________________________________________________

59.9 problem number 9

problem number 544

Added Feb. 1, 2019.

Problem 2.7.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (arctan x)^n y^2 - b \lambda x^m (\arctan x)^n y+ b m x^{m-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 - b*lambda*x^m*ArcTan[x]^n*y + b*m*x^(m - 1))*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+(lambda*arctan(x)^n*y^2 - b*lambda*x^m*arctan(x)^n*y+ b*m*x^(m-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

59.10 problem number 10

problem number 545

Added Feb. 1, 2019.

Problem 2.7.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (arctan x)^n y^2 +b m x^{m-1} - \lambda b^2 x^{2 m}(\arctan x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcTan[x]^n)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+(lambda*arctan(x)^n*y^2 +b*m*x^(m-1) - lambda*b^2*x^(2*m)*arctan(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

59.11 problem number 11

problem number 546

Added Feb. 1, 2019.

Problem 2.7.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (arctan x)^n (y-a x^m -b)^2 + a m x^{m-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = D[w[x, y], x] + (lambda*ArcTan[x]^n*(y - a*x^m - b)^2 + a*m*x^(m - 1))*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  diff(w(x,y),x)+(lambda*arctan(x)^n*(y-a*x^m -b)^2 + a*m*x^(m-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{x}^{m}\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda \,{\rm d}xa+y\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda \,{\rm d}x-\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda \,{\rm d}xb+1}{y-a{x}^{m}-b}} \right ) \]

____________________________________________________________________________________

59.12 problem number 12

problem number 547

Added Feb. 1, 2019.

Problem 2.7.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( \lambda (\arctan x)^n y^2+k y+ \lambda b^2 x^{2 k} (\arctan x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; 
 pde = x*D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcTan[x]^n)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde :=  x*diff(w(x,y),x)+(lambda*arctan(x)^n*y^2+k*y+lambda*b^2*x^(2*k)*arctan(x)^n  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \lambda \,b\int \! \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{k-1}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]