118 HFOPDE, chapter 4.7.2

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118.1 Problem 1

problem number 954

Added March 9, 2019.

Problem Chapter 4.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c \arccos (\frac {x}{\lambda } + k \arccos (\frac {y}{\beta } ) \right ) w \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcCos[x/lambda] + k*ArcCos[y/beta])*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {-\frac {k \left (a^2 \left (\beta ^2-y^2\right )+i \sqrt {a^2 \left (\beta ^2-y^2\right )} (a y-b x) \log \left (2 \left (\sqrt {a^2 \left (\beta ^2-y^2\right )}-i a y\right )\right )\right )}{b \beta \sqrt {1-\frac {y^2}{\beta ^2}}}+a k x \cos ^{-1}\left (\frac {y}{\beta }\right )-a c \lambda \sqrt {1-\frac {x^2}{\lambda ^2}}+a c x \cos ^{-1}\left (\frac {x}{\lambda }\right )}{a^2}\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arccos(x/lambda)+k*arccos(y/beta))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {1}{ab} \left ( -\sqrt {{\frac {{\lambda }^{2}-{x}^{2}}{{\lambda }^{2}}}}bc\lambda -\sqrt {{\frac {{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}a\beta \,k+\arccos \left ( {\frac {y}{\beta }} \right ) aky+\arccos \left ( {\frac {x}{\lambda }} \right ) bcx \right ) }}} \]

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118.2 Problem 2

problem number 955

Added March 9, 2019.

Problem Chapter 4.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcCos[lambda*x + beta*y]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left (\beta (b x-a y) \sin ^{-1}(\beta y+\lambda x)+x (a \lambda +b \beta ) \cos ^{-1}(\beta y+\lambda x)+a \left (-\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}\right )\right )}{a (a \lambda +b \beta )}\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arccos(lambda*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac { \left ( -\sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}+\arccos \left ( \beta \,y+\lambda \,x \right ) \left ( \beta \,y+\lambda \,x \right ) \right ) c}{a\lambda +b\beta }}}} \]

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118.3 Problem 3

problem number 956

Added March 9, 2019.

Problem Chapter 4.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcCos[lambda*x + beta*y]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {\left (a^2+2 \beta ^2 (b x-a y)^2\right ) \sin ^{-1}(\beta y+\lambda x)-a \sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1} (-3 a \beta y+a \lambda x+4 b \beta x)+2 x^2 (a \lambda +b \beta )^2 \cos ^{-1}(\beta y+\lambda x)}{4 (a \lambda +b \beta )^2}\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arccos(lambda*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{1/4\,{\frac {a \left ( -2\,\arccos \left ( \beta \,y+\lambda \,x \right ) a{\beta }^{2}{y}^{2}+2\,\arccos \left ( \beta \,y+\lambda \,x \right ) a{\lambda }^{2}{x}^{2}+4\,\arccos \left ( \beta \,y+\lambda \,x \right ) b{\beta }^{2}xy+4\,\arccos \left ( \beta \,y+\lambda \,x \right ) b\beta \,\lambda \,{x}^{2}+3\,\sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}a\beta \,y-\sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}a\lambda \,x-4\,\sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}b\beta \,x+\arcsin \left ( \beta \,y+\lambda \,x \right ) a \right ) }{ \left ( a\lambda +b\beta \right ) ^{2}}}}} \]

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118.4 Problem 4

problem number 957

Added March 9, 2019.

Problem Chapter 4.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*D[w[x, y], x] + b*ArcCos[lambda*x]^n*D[w[x, y], y] == (c*ArcCos[mu*x]^m + s*ArcCos[beta*y]^k)*w[x, y]; 
 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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*diff(w(x,y),x)+ b*arccos(lambda*x)^n*diff(w(x,y),y) =(c*arccos(mu*x)^m+s*arccos(beta*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( \left ( n+2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) -\arccos \left ( \lambda \,x \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) + \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+3/2} \right ) b\sqrt {-{\lambda }^{2}{x}^{2}+1}+\lambda \, \left ( n+2 \right ) \left ( -\arccos \left ( \lambda \,x \right ) bx\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) +a\sqrt {\arccos \left ( \lambda \,x \right ) }y \right ) }{a\lambda \, \left ( n+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \arccos \left ( {\it \_b}\,\mu \right ) \right ) ^{m}+s \left ( \arccos \left ( {\frac { \left ( \left ( \left ( -n-2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) +\arccos \left ( {\it \_b}\,\lambda \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) - \left ( \arccos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n+3/2} \right ) b\sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1}+ \left ( n+2 \right ) \left ( \arccos \left ( {\it \_b}\,\lambda \right ) b{\it \_b}\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) +\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) } \left ( ya-b\int \! \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x \right ) \right ) \lambda \right ) \beta }{a\lambda \, \left ( n+2 \right ) \sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }}} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]

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118.5 Problem 5

problem number 958

Added March 9, 2019.

Problem Chapter 4.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*D[w[x, y], x] + b*ArcCos[lambda*y]^n*D[w[x, y], y] == (c*ArcCos[mu*x]^m + s*ArcCos[beta*y]^k)*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \] 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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*diff(w(x,y),x)+ b*arccos(lambda*y)^n*diff(w(x,y),y) =(c*arccos(mu*x)^m+s*arccos(beta*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{2}^{-n}a\sqrt {\pi }}{\lambda \,b} \left ( -{\frac { \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n+1}{2}^{n}\sqrt {-{y}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+{\frac {{2}^{n}\sqrt {\arccos \left ( y\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( y\lambda \right ) \right ) \sqrt {-{y}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+3/2\,{\frac {{2}^{n} \left ( -2/3\,n+4/3 \right ) \left ( \arccos \left ( y\lambda \right ) y\lambda -\sqrt {-{y}^{2}{\lambda }^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( y\lambda \right ) \right ) }{\sqrt {\pi } \left ( n-2 \right ) \sqrt {\arccos \left ( y\lambda \right ) }}} \right ) }+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \arccos \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \arccos \left ( \mu \, \left ( -{\frac {{2}^{-n}a\sqrt {\pi }}{\lambda \,b} \left ( -{\frac { \left ( \arccos \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n+1}{2}^{n}\sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+{\frac {{2}^{n}\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) \sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+3/2\,{\frac {{2}^{n} \left ( -2/3\,n+4/3 \right ) \left ( \arccos \left ( {\it \_b}\,\lambda \right ) {\it \_b}\,\lambda -\sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) }{\sqrt {\pi } \left ( n-2 \right ) \sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }}} \right ) }-{\frac {a\int \! \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \right ) ^{m}+s \left ( \arccos \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]