117 HFOPDE, chapter 4.7.1

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117.1 Problem 1

problem number 949

Added March 9, 2019.

Problem Chapter 4.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcSin[x/lambda] + k*ArcSin[y/beta])*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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*arcsin(x/lambda)+k*arcsin(y/beta))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

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

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117.2 Problem 2

problem number 950

Added March 9, 2019.

Problem Chapter 4.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcSin[lambda*x + beta*y]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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*arcsin(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);
 

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

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117.3 Problem 3

problem number 951

Added March 9, 2019.

Problem Chapter 4.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcSin[lambda*x + beta*y]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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*arcsin(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);
 

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

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117.4 Problem 4

problem number 952

Added March 9, 2019.

Problem Chapter 4.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcSin[lambda*x]^n*D[w[x, y], y] == (c*ArcSin[mu*x]^m + s*ArcSin[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*arcsin(lambda*x)^n*diff(w(x,y),y) =(c*arcsin(mu*x)^m+s*arcsin(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);
 

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

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117.5 Problem 5

problem number 953

Added March 9, 2019.

Problem Chapter 4.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcSin[lambda*y]^n*D[w[x, y], y] == (c*ArcSin[mu*x]^m + s*ArcSin[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*arcsin(lambda*y)^n*diff(w(x,y),y) =(c*arcsin(mu*x)^m+s*arcsin(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);
 

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