6.5.19 7.1

6.5.19.1 [1323] Problem 1

problem number 1323

Added April 13, 2019.

Problem Chapter 5.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {\textit {\_a}}{a}} \left (\operatorname {c2} \arcsin \left (\frac {\beta \left (a y -b \left (x -\textit {\_a} \right )\right )}{a}\right )^{n}+\operatorname {c1} \arcsin \left (\lambda \textit {\_a} \right )^{k}\right )d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]

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6.5.19.2 [1324] Problem 2

problem number 1324

Added April 13, 2019.

Problem Chapter 5.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.19.3 [1325] Problem 3

problem number 1325

Added April 13, 2019.

Problem Chapter 5.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.19.4 [1326] Problem 4

problem number 1326

Added April 13, 2019.

Problem Chapter 5.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {{\mathrm e}^{\frac {c \int \arcsin \left (\nu x \right )^{k}d x}{a}} \left (p \int _{}^{x}{\arcsin \left (\frac {\left (\mu \left (\operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu \textit {\_f} \right )\right ) \textit {\_f} b +\arcsin \left (\mu \textit {\_f} \right ) \operatorname {LommelS1}\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu \textit {\_f} \right )\right ) b m \textit {\_f} +\sqrt {\arcsin \left (\mu \textit {\_f} \right )}\, \left (m +1\right ) \left (-b \int \arcsin \left (\mu x \right )^{m}d x +y a \right )\right ) \sqrt {-\textit {\_f}^{2} \mu ^{2}+1}+b \left (\mu \textit {\_f} -1\right ) \left (\mu \textit {\_f} +1\right ) \left (\operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu \textit {\_f} \right )\right ) \arcsin \left (\mu \textit {\_f} \right )-\arcsin \left (\mu \textit {\_f} \right )^{m +\frac {3}{2}}\right )\right ) \beta }{\sqrt {-\textit {\_f}^{2} \mu ^{2}+1}\, \sqrt {\arcsin \left (\mu \textit {\_f} \right )}\, \left (m +1\right ) \mu a}\right )}^{n} {\mathrm e}^{\frac {\left (-\nu \textit {\_f} \left (\operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\nu \textit {\_f} \right )\right ) k \arcsin \left (\nu \textit {\_f} \right )+\operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\nu \textit {\_f} \right )\right )\right ) \sqrt {-\textit {\_f}^{2} \nu ^{2}+1}+\left (\nu \textit {\_f} -1\right ) \left (\nu \textit {\_f} +1\right ) \left (-\arcsin \left (\nu \textit {\_f} \right ) \operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\nu \textit {\_f} \right )\right )+\arcsin \left (\nu \textit {\_f} \right )^{k +\frac {3}{2}}\right )\right ) c}{\sqrt {-\textit {\_f}^{2} \nu ^{2}+1}\, \sqrt {\arcsin \left (\nu \textit {\_f} \right )}\, a \nu \left (1+k \right )}}d \textit {\_f} +f_{1} \left (\frac {b \left (\sqrt {\arcsin \left (\mu x \right )}\, \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right )-\arcsin \left (\mu x \right )^{m +1}\right ) \sqrt {-\mu ^{2} x^{2}+1}+\left (-\frac {b x \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right )}{\sqrt {\arcsin \left (\mu x \right )}}-b m x \sqrt {\arcsin \left (\mu x \right )}\, \operatorname {LommelS1}\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu x \right )\right )+a y \left (m +1\right )\right ) \mu }{\left (m +1\right ) a \mu }\right ) a \right )}{a}\]

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6.5.19.5 [1327] Problem 5

problem number 1327

Added April 13, 2019.

Problem Chapter 5.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {\left (p \int _{}^{x}\arcsin \left (\beta \textit {\_b} \right )^{n} {\mathrm e}^{-\frac {c \int {\arcsin \left (\frac {\left (\left (\operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu \textit {\_b} \right )\right ) \textit {\_b} b +\arcsin \left (\mu \textit {\_b} \right ) \operatorname {LommelS1}\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu \textit {\_b} \right )\right ) b m \textit {\_b} +\sqrt {\arcsin \left (\mu \textit {\_b} \right )}\, \left (m +1\right ) \left (y a -b \int \arcsin \left (\mu x \right )^{m}d x \right )\right ) \mu \sqrt {-\textit {\_b}^{2} \mu ^{2}+1}+b \left (\mu \textit {\_b} -1\right ) \left (\mu \textit {\_b} +1\right ) \left (\arcsin \left (\mu \textit {\_b} \right ) \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu \textit {\_b} \right )\right )-\arcsin \left (\mu \textit {\_b} \right )^{m +\frac {3}{2}}\right )\right ) \nu }{\sqrt {\arcsin \left (\mu \textit {\_b} \right )}\, \sqrt {-\textit {\_b}^{2} \mu ^{2}+1}\, \left (m +1\right ) a \mu }\right )}^{k}d \textit {\_b}}{a}}d \textit {\_b} +f_{1} \left (\frac {b \left (\sqrt {\arcsin \left (\mu x \right )}\, \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right )-\arcsin \left (\mu x \right )^{m +1}\right ) \sqrt {-\mu ^{2} x^{2}+1}+\left (-\frac {b x \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right )}{\sqrt {\arcsin \left (\mu x \right )}}-b m x \sqrt {\arcsin \left (\mu x \right )}\, \operatorname {LommelS1}\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu x \right )\right )+a y \left (m +1\right )\right ) \mu }{\left (m +1\right ) a \mu }\right ) a \right ) {\mathrm e}^{\frac {c \int _{}^{x}{\arcsin \left (\frac {\left (\left (\operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu \textit {\_b} \right )\right ) \textit {\_b} b +\arcsin \left (\mu \textit {\_b} \right ) \operatorname {LommelS1}\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu \textit {\_b} \right )\right ) b m \textit {\_b} +\sqrt {\arcsin \left (\mu \textit {\_b} \right )}\, \left (m +1\right ) \left (y a -b \int \arcsin \left (\mu x \right )^{m}d x \right )\right ) \mu \sqrt {-\textit {\_b}^{2} \mu ^{2}+1}+b \left (\mu \textit {\_b} -1\right ) \left (\mu \textit {\_b} +1\right ) \left (\arcsin \left (\mu \textit {\_b} \right ) \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu \textit {\_b} \right )\right )-\arcsin \left (\mu \textit {\_b} \right )^{m +\frac {3}{2}}\right )\right ) \nu }{\sqrt {\arcsin \left (\mu \textit {\_b} \right )}\, \sqrt {-\textit {\_b}^{2} \mu ^{2}+1}\, \left (m +1\right ) a \mu }\right )}^{k}d \textit {\_b}}{a}}}{a}\]

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