6.2.21 7.2

6.2.21.1 [715] problem number 1

problem number 715

Added January 29, 2019.

Problem 2.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcCos[lambda*x]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a \arccos (\lambda x)^k (-i \arccos (\lambda x))^{-k} \Gamma (k+1,-i \arccos (\lambda x))+a (i \arccos (\lambda x))^{-k} \arccos (\lambda x)^k \Gamma (k+1,i \arccos (\lambda x))+2 b \lambda x-2 \lambda y}{2 \lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arccos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (\left (2+k \right ) \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )-\operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \arccos \left (\lambda x \right )+\arccos \left (\lambda x \right )^{k +\frac {3}{2}}\right ) a \sqrt {-\lambda ^{2} x^{2}+1}-\lambda \left (a \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) x \arccos \left (\lambda x \right )+\sqrt {\arccos \left (\lambda x \right )}\, \left (b x -y \right )\right ) \left (2+k \right )}{\sqrt {\arccos \left (\lambda x \right )}\, \lambda \left (2+k \right )}\right )\]

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6.2.21.2 [716] problem number 2

problem number 716

Added January 29, 2019.

Problem 2.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcCos[lambda*y]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arccos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\int \frac {1}{a \arccos \left (\lambda y \right )^{k}+b}d y +x \right )\]

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6.2.21.3 [717] problem number 3

problem number 717

Added January 29, 2019.

Problem 2.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {b c_1}{b k \arccos (c+a x+b K[6006])^n+a}-\int _1^x-\frac {a b^2 k n \arccos (c+a K[1]+b K[6006])^{n-1} c_1}{\left (b k \arccos (c+a K[1]+b K[6006])^n+a\right )^2 \sqrt {1-(c+a K[1]+b K[6006])^2}}dK[1]\right )dK[6006]+\int _1^x\frac {b k \arccos (c+b y+a K[1])^n c_1}{b k \arccos (c+b y+a K[1])^n+a}dK[1]+c_2\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+k*arccos(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'));
 

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

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6.2.21.4 [718] problem number 4

problem number 718

Added January 29, 2019.

Problem 2.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*ArcCos[lambda*x]^k*ArcCos[mu*y]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\left (\arccos (\lambda x)^2\right )^{-k} \left (-a (i \arccos (\lambda x))^k \arccos (\lambda x)^k \Gamma (k+1,-i \arccos (\lambda x))-a (-i \arccos (\lambda x))^k \arccos (\lambda x)^k \Gamma (k+1,i \arccos (\lambda x))+\frac {\lambda \left (\arccos (\lambda x)^2\right )^k \arccos (\mu y)^{-n} \left ((-i \arccos (\mu y))^n \Gamma (1-n,-i \arccos (\mu y))+(i \arccos (\mu y))^n \Gamma (1-n,i \arccos (\mu y))\right )}{\mu }\right )}{2 \lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a*arccos(lambda*x)^k*arccos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (\left (2+k \right ) \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )-\operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \arccos \left (\lambda x \right )+\arccos \left (\lambda x \right )^{k +\frac {3}{2}}\right ) \left (-2+n \right ) a \mu \sqrt {\arccos \left (\mu y \right )}\, \sqrt {-\lambda ^{2} x^{2}+1}-\left (-\sqrt {\arccos \left (\lambda x \right )}\, \left (\left (2-n \right ) \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu y \right )\right )-\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mu y \right )\right ) \arccos \left (\mu y \right )+\arccos \left (\mu y \right )^{-n +\frac {3}{2}}\right ) \sqrt {-\mu ^{2} y^{2}+1}+\mu \left (-2+n \right ) \left (a \sqrt {\arccos \left (\mu y \right )}\, \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) x \arccos \left (\lambda x \right )-\sqrt {\arccos \left (\lambda x \right )}\, y \arccos \left (\mu y \right ) \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu y \right )\right )\right )\right ) \lambda \left (2+k \right )}{\sqrt {\arccos \left (\lambda x \right )}\, \sqrt {\arccos \left (\mu y \right )}\, a \mu \left (-2+n \right ) \lambda \left (2+k \right )}\right )\]

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6.2.21.5 [719] problem number 5

problem number 719

Added January 29, 2019.

Problem 2.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*ArcCos[x]^n*y - a^2 + a*lambda*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( y^2+lambda*arccos(x)^n*y- a^2 + a*lambda*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

time expired

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6.2.21.6 [720] problem number 6

problem number 720

Added January 29, 2019.

Problem 2.7.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*x*ArcCos[x]^n*y + a*lambda*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( y^2+lambda*x*arccos(x)^n*y + a*lambda*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.21.7 [721] problem number 7

problem number 721

Added January 29, 2019.

Problem 2.7.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcCos[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]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)-( (k+1)*x^k*y^2 -lambda*arccos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {x^{k +1} {\mathrm e}^{\int \frac {x^{k +1} \arccos \left (x \right )^{n} \lambda x -2 k -2}{x}d x}-\int x^{-k -2} {\mathrm e}^{\lambda \int x^{k +1} \arccos \left (x \right )^{n}d x}d x \left (x^{k +1} y -1\right ) \left (k +1\right )}{x^{k +1} y -1}\right )\]

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6.2.21.8 [722] problem number 8

problem number 722

Added January 29, 2019.

Problem 2.7.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + a*y + a*b - b^2*lambda*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ a*y+ a*b - b^2*lambda*arccos(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 ) = f_{1} \left (\frac {-\int \frac {4 \,{\mathrm e}^{\frac {2 b \left (\left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\arccos \left (x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \lambda \sqrt {-x^{2}+1}+x \left (2+n \right ) \left (-2 b \lambda \arccos \left (x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )+a \sqrt {\arccos \left (x \right )}\right )}{\sqrt {\arccos \left (x \right )}\, \left (2+n \right )}} \left (\left (\frac {n b \left (\pi -2 \arcsin \left (x \right )\right )^{2} \left (2+n \right ) \operatorname {LommelS1}\left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )}{2}-2 \arccos \left (x \right ) b \left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )+\frac {b \left (\pi -2 \arcsin \left (x \right )\right )^{2} \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right )}{2}+\left (\left (b +y \right ) n +2 y \right ) \arccos \left (x \right )^{n +\frac {5}{2}}\right ) \sqrt {-x^{2}+1}+2 x b \left (-\frac {n \left (\pi ^{3}-12 \arccos \left (x \right ) \pi \arcsin \left (x \right )-8 \arcsin \left (x \right )^{3}\right ) \left (2+n \right ) \operatorname {LommelS1}\left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )}{8}+\frac {\left (\pi -2 \arcsin \left (x \right )\right )^{2} \left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )}{4}+\left (-\frac {\pi ^{3}}{8}+\frac {3 \arccos \left (x \right ) \pi \arcsin \left (x \right )}{2}+\arcsin \left (x \right )^{3}\right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right )+\arccos \left (x \right )^{\frac {7}{2}+n}\right )\right )}{\arccos \left (x \right )^{{5}/{2}} \sqrt {-x^{2}+1}}d x \lambda -4 \,{\mathrm e}^{\frac {2 b \left (\left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\arccos \left (x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \lambda \sqrt {-x^{2}+1}+x \left (2+n \right ) \left (-2 b \lambda \arccos \left (x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )+a \sqrt {\arccos \left (x \right )}\right )}{\sqrt {\arccos \left (x \right )}\, \left (2+n \right )}} \left (2+n \right )}{4 \left (2+n \right ) \left (b +y \right )}\right )\]

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6.2.21.9 [723] problem number 9

problem number 723

Added January 29, 2019.

Problem 2.7.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 - b*lambda*x^m*ArcCos[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]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(x)^n*y^2- b*lambda*x^m*arccos(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'));
 

sol=()

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6.2.21.10 [724] problem number 10

problem number 724

Added January 29, 2019.

Problem 2.7.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ b*m*x^(m-1) - lambda*b^2*x^(2*m)*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.21.11 [725] problem number 11

problem number 725

Added January 29, 2019.

Problem 2.7.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \left (-\frac {2}{a x^m+b-y}+\lambda (i \arccos (x))^n \arccos (x)^n \left (\arccos (x)^2\right )^{-n} \Gamma (n+1,-i \arccos (x))+\lambda (-i \arccos (x))^n \arccos (x)^n \left (\arccos (x)^2\right )^{-n} \Gamma (n+1,i \arccos (x))\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (\left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )+\arccos \left (x \right )^{n} \arccos \left (x \right )^{{3}/{2}}-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )\right ) \left (a \,x^{m}+b -y \right ) \lambda \sqrt {-x^{2}+1}-\left (2+n \right ) \left (x \lambda \arccos \left (x \right ) \left (a \,x^{m}+b -y \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\sqrt {\arccos \left (x \right )}\right )}{\sqrt {\arccos \left (x \right )}\, \left (2+n \right ) \left (a \,x^{m}+b -y \right )}\right )\]

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6.2.21.12 [726] problem number 12

problem number 726

Added January 29, 2019.

Problem 2.7.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x\lambda \arccos (K[1])^n K[1]^{k-1}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  x*diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ k*y + lambda*b^2*x^(2*k)*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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