7.5.20 7.2

7.5.20.1 [1330] Problem 1

problem number 1330

Added April 13, 2019.

Problem Chapter 5.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \arccos ^k(\lambda x) + c_2 \arccos ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcCos[lambda*x]^k+c2*ArcCos[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \cos ^{-1}(\lambda K[1])^k+\text {c2} \cos ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  w(x,y)+c1*arccos(lambda*x)^k+c2*arccos(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \arccos \left (\textit {\_a} \lambda \right )^{k}+\mathit {c2} \arccos \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n}\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]

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7.5.20.2 [1331] Problem 2

problem number 1331

Added April 13, 2019.

Problem Chapter 5.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\arccos \left (\textit {\_a} \lambda \right )^{k} \arccos \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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7.5.20.3 [1332] Problem 3

problem number 1332

Added April 13, 2019.

Problem Chapter 5.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c_1 \arccos (\lambda _1 x) + c_2 \arccos (\lambda _2 y)\right ) w+ s_1 \arccos ^n(\beta _1 x)+ s_2 \arccos ^k(\beta _2 y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcCos[lambda1*x] + c2*ArcCos[lambda2*y])*w[x,y]+ s1*ArcCos[beta1*x]^n+ s2*ArcCos[beta2*y]^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (-\frac {\text {c1} \sqrt {1-\text {lambda1}^2 x^2}}{a \text {lambda1}}+\frac {\text {c1} x \cos ^{-1}(\text {lambda1} x)}{a}+\frac {\text {c2} x \sin ^{-1}(\text {lambda2} y)}{a}+\frac {\text {c2} x \cos ^{-1}(\text {lambda2} y)}{a}-\frac {\text {c2} \sqrt {1-\text {lambda2}^2 y^2}}{b \text {lambda2}}-\frac {\text {c2} y \sin ^{-1}(\text {lambda2} y)}{b}\right ) \left (\int _1^x\frac {\exp \left (\frac {\text {c2} \text {lambda2} (a y-b x) \sin ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \text {lambda1}-b \text {c1} \text {lambda2} \cos ^{-1}(\text {lambda1} K[1]) K[1] \text {lambda1}-b \text {c2} \text {lambda2} \cos ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) K[1] \text {lambda1}+a \text {c2} \sqrt {-y^2 \text {lambda2}^2-\frac {b^2 (x-K[1])^2 \text {lambda2}^2}{a^2}+\frac {2 b y (x-K[1]) \text {lambda2}^2}{a}+1} \text {lambda1}+b \text {c1} \text {lambda2} \sqrt {1-\text {lambda1}^2 K[1]^2}}{a b \text {lambda1} \text {lambda2}}\right ) \left (\text {s2} \cos ^{-1}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \cos ^{-1}(\text {beta1} K[1])^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arccos(lambda1*x) + c2*arccos(lambda2*y))*w(x,y)+ s1*arccos(beta1*x)^n+ s2*arccos(beta2*y)^k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {s1} \arccos \left (\textit {\_a} \beta 1 \right )^{n}+\mathit {s2} \arccos \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta 2}{a}\right )^{k}\right ) {\mathrm e}^{\frac {\sqrt {-\frac {\left (-\left (-\textit {\_a} +x \right ) b \lambda 2 +\left (\lambda 2 y -1\right ) a \right ) \left (-\left (-\textit {\_a} +x \right ) b \lambda 2 +\left (\lambda 2 y +1\right ) a \right )}{a^{2}}}\, a \mathit {c2} \lambda 1 -\left (\left (a y +\left (\textit {\_a} -x \right ) b \right ) \mathit {c2} \lambda 1 \arccos \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )+\left (\textit {\_a} \lambda 1 \arccos \left (\textit {\_a} \lambda 1 \right )-\sqrt {-\textit {\_a}^{2} \lambda 1^{2}+1}\right ) b \mathit {c1} \right ) \lambda 2}{a b \lambda 1 \lambda 2}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {-\sqrt {-\lambda 2^{2} y^{2}+1}\, a \mathit {c2} \lambda 1 +\left (-\sqrt {-\lambda 1^{2} x^{2}+1}\, b \mathit {c1} +\left (a \mathit {c2} y \arccos \left (\lambda 2 y \right )+b \mathit {c1} x \arccos \left (\lambda 1 x \right )\right ) \lambda 1 \right ) \lambda 2}{a b \lambda 1 \lambda 2}}\]

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7.5.20.4 [1333] Problem 4

problem number 1333

Added April 13, 2019.

Problem Chapter 5.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu x) w + p \arccos ^n(\beta y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c \cos ^{-1}(\nu x)^k \left (\cos ^{-1}(\nu x)^2\right )^{-k} \left (\left (-i \cos ^{-1}(\nu x)\right )^k \operatorname {Gamma}\left (k+1,i \cos ^{-1}(\nu x)\right )+\left (i \cos ^{-1}(\nu x)\right )^k \operatorname {Gamma}\left (k+1,-i \cos ^{-1}(\nu x)\right )\right )}{2 a \nu }\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \cos ^{-1}(\nu K[2])^k \left (\cos ^{-1}(\nu K[2])^2\right )^{-k} \left (\operatorname {Gamma}\left (k+1,i \cos ^{-1}(\nu K[2])\right ) \left (-i \cos ^{-1}(\nu K[2])\right )^k+\left (i \cos ^{-1}(\nu K[2])\right )^k \operatorname {Gamma}\left (k+1,-i \cos ^{-1}(\nu K[2])\right )\right )}{2 a \nu }\right ) p \cos ^{-1}\left (\beta \left (y-\int _1^x\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^n}{a}dK[2]+c_1\left (y-\int _1^x\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \arccos \left (\frac {\left (\frac {\left (-\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mu x \right )\right ) \arccos \left (\mu x \right )+\arccos \left (\mu x \right )^{m +\frac {3}{2}}+\left (m +2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right )\right ) \sqrt {-\mu ^{2} x^{2}+1}\, b \sqrt {\arccos \left (\textit {\_a} \mu \right )}}{\sqrt {\arccos \left (\mu x \right )}}+\left (\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_a} \mu \right )\right ) \arccos \left (\textit {\_a} \mu \right )-\arccos \left (\textit {\_a} \mu \right )^{m +\frac {3}{2}}+\left (-m -2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \mu \right )\right )\right ) \sqrt {-\textit {\_a}^{2} \mu ^{2}+1}\, b +\left (m +2\right ) \left (\textit {\_a} b \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \mu \right )\right ) \arccos \left (\textit {\_a} \mu \right )-b x \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right ) \sqrt {\arccos \left (\textit {\_a} \mu \right )}\, \sqrt {\arccos \left (\mu x \right )}+a y \sqrt {\arccos \left (\textit {\_a} \mu \right )}\right ) \mu \right ) \beta }{\left (m +2\right ) a \mu \sqrt {\arccos \left (\textit {\_a} \mu \right )}}\right )^{n} {\mathrm e}^{-\frac {\left (\left (k +2\right ) \textit {\_a} \nu \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \nu \right )\right ) \arccos \left (\textit {\_a} \nu \right )+\left (\LommelS 1 \left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_a} \nu \right )\right ) \arccos \left (\textit {\_a} \nu \right )-\arccos \left (\textit {\_a} \nu \right )^{k +\frac {3}{2}}+\left (-k -2\right ) \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \nu \right )\right )\right ) \sqrt {-\textit {\_a}^{2} \nu ^{2}+1}\right ) c}{\left (k +2\right ) a \nu \sqrt {\arccos \left (\textit {\_a} \nu \right )}}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {\left (-\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mu x \right )\right ) \arccos \left (\mu x \right )+\arccos \left (\mu x \right )^{m +\frac {3}{2}}+\left (m +2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right )\right ) \sqrt {-\mu ^{2} x^{2}+1}\, b +\left (m +2\right ) \left (-b x \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right ) \arccos \left (\mu x \right )+a y \sqrt {\arccos \left (\mu x \right )}\right ) \mu }{\left (m +2\right ) a \mu \sqrt {\arccos \left (\mu x \right )}}\right )\right ) {\mathrm e}^{-\frac {\left (-\left (k +2\right ) \nu x \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\nu x \right )\right ) \arccos \left (\nu x \right )+\left (-\LommelS 1 \left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\nu x \right )\right ) \arccos \left (\nu x \right )+\arccos \left (\nu x \right )^{k +\frac {3}{2}}+\left (k +2\right ) \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\nu x \right )\right )\right ) \sqrt {-\nu ^{2} x^{2}+1}\right ) c 2^{k} 2^{-k}}{\left (k +2\right ) a \nu \sqrt {\arccos \left (\nu x \right )}}}\]

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7.5.20.5 [1334] Problem 5

problem number 1334

Added April 13, 2019.

Problem Chapter 5.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu y) w + p \arccos ^n(\beta x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cos ^{-1}\left (\nu \left (y-\int _1^x\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cos ^{-1}\left (\nu \left (y-\int _1^x\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) p \cos ^{-1}(\beta K[3])^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cos ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \arccos \left (\textit {\_b} \beta \right )^{n} {\mathrm e}^{-\frac {c \left (\int \arccos \left (\frac {\left (\frac {\left (-\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mu x \right )\right ) \arccos \left (\mu x \right )+\arccos \left (\mu x \right )^{m +\frac {3}{2}}+\left (m +2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right )\right ) \sqrt {-\mu ^{2} x^{2}+1}\, b \sqrt {\arccos \left (\textit {\_b} \mu \right )}}{\sqrt {\arccos \left (\mu x \right )}}+\left (\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_b} \mu \right )\right ) \arccos \left (\textit {\_b} \mu \right )-\arccos \left (\textit {\_b} \mu \right )^{m +\frac {3}{2}}+\left (-m -2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_b} \mu \right )\right )\right ) \sqrt {-\textit {\_b}^{2} \mu ^{2}+1}\, b +\left (m +2\right ) \left (\textit {\_b} b \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_b} \mu \right )\right ) \arccos \left (\textit {\_b} \mu \right )-b x \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right ) \sqrt {\arccos \left (\textit {\_b} \mu \right )}\, \sqrt {\arccos \left (\mu x \right )}+a y \sqrt {\arccos \left (\textit {\_b} \mu \right )}\right ) \mu \right ) \nu }{\left (m +2\right ) a \mu \sqrt {\arccos \left (\textit {\_b} \mu \right )}}\right )^{k}d \textit {\_b} \right )}{a}}}{a}d \textit {\_b} +\textit {\_F1} \left (\frac {\left (-\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mu x \right )\right ) \arccos \left (\mu x \right )+\arccos \left (\mu x \right )^{m +\frac {3}{2}}+\left (m +2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right )\right ) \sqrt {-\mu ^{2} x^{2}+1}\, b +\left (m +2\right ) \left (-b x \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right ) \arccos \left (\mu x \right )+a y \sqrt {\arccos \left (\mu x \right )}\right ) \mu }{\left (m +2\right ) a \mu \sqrt {\arccos \left (\mu x \right )}}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \arccos \left (\frac {\left (-\left (-\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_a} \mu \right )\right ) \sqrt {\arccos \left (\textit {\_a} \mu \right )}+\arccos \left (\textit {\_a} \mu \right )^{m +1}+\frac {\left (m +2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \mu \right )\right )}{\sqrt {\arccos \left (\textit {\_a} \mu \right )}}\right ) \sqrt {-\textit {\_a}^{2} \mu ^{2}+1}\, b +\left (m +2\right ) \left (\textit {\_a} b \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \mu \right )\right ) \sqrt {\arccos \left (\textit {\_a} \mu \right )}-b x \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right ) \sqrt {\arccos \left (\mu x \right )}+a y \right ) \mu +\frac {\left (-\LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mu x \right )\right ) \arccos \left (\mu x \right )+\arccos \left (\mu x \right )^{m +\frac {3}{2}}+\left (m +2\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu x \right )\right )\right ) \sqrt {-\mu ^{2} x^{2}+1}\, b}{\sqrt {\arccos \left (\mu x \right )}}\right ) \nu }{\left (m +2\right ) a \mu }\right )^{k}}{a}d \textit {\_a}}\]

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