116 HFOPDE, chapter 4.6.5

____________________________________________________________________________________

116.1 Problem 1

problem number 943

Added March 9, 2019.

Problem Chapter 4.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = (b \sin (\lambda x)+k \cos (\mu y)) w \]

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 = D[w[x, y], x] + a*D[w[x, y], y] == (b*Sin[lambda*x] + k*Cos[mu*y])*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to c_1(y-a x) e^{\frac {k \sin (\mu y)}{a \mu }-\frac {b \cos (\lambda x)}{\lambda }}\right \}\right \} \]

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 :=  diff(w(x,y),x)+ a*diff(w(x,y),y) = (b*sin(lambda*x)+k*cos(mu*y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ) {{\rm e}^{{\frac {-b\cos \left ( \lambda \,x \right ) \mu \,a+k\sin \left ( \mu \,y \right ) \lambda }{\lambda \,\mu \,a}}}} \]

____________________________________________________________________________________

116.2 Problem 2

problem number 944

Added March 9, 2019.

Problem Chapter 4.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \sin (\mu y) w_y = (b \sin (\lambda x)+k \tan (\mu y)) w \]

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 = D[w[x, y], x] + a*D[w[x, y], y] == (b*Sin[lambda*x] + k*Tan[mu*y])*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to c_1(y-a x) e^{-\frac {b \cos (\lambda x)}{\lambda }} \cos ^{-\frac {k}{a \mu }}(\mu y)\right \}\right \} \]

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 :=  diff(w(x,y),x)+ a*diff(w(x,y),y) = (b*sin(lambda*x)+k*tan(mu*y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ) \left ( \left ( \cos \left ( 1/2\,\mu \,y \right ) \right ) ^{-2} \right ) ^{{\frac {k}{\mu \,a}}} \left ( {\frac {\sin \left ( 1/2\,\mu \,y \right ) -\cos \left ( 1/2\,\mu \,y \right ) }{\cos \left ( 1/2\,\mu \,y \right ) }} \right ) ^{-{\frac {k}{\mu \,a}}} \left ( {\frac {\sin \left ( 1/2\,\mu \,y \right ) +\cos \left ( 1/2\,\mu \,y \right ) }{\cos \left ( 1/2\,\mu \,y \right ) }} \right ) ^{-{\frac {k}{\mu \,a}}}{{\rm e}^{-2\,{\frac {b \left ( \cos \left ( 1/2\,\lambda \,x \right ) \right ) ^{2}}{\lambda }}}} \]

____________________________________________________________________________________

116.3 Problem 3

problem number 945

Added March 9, 2019.

Problem Chapter 4.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \sin (\mu y) w_y = b \tan (\lambda x) w \]

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 = D[w[x, y], x] + a*Sin[mu*y]*D[w[x, y], y] == b*Tan[lambda*x]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \cos ^{-\frac {b}{\lambda }}(\lambda x) c_1\left (\frac {\log \left (\tan \left (\frac {\mu y}{2}\right )\right )-a \mu x}{\mu }\right )\right \}\right \} \]

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 :=  diff(w(x,y),x)+ a*sin(mu*y)*diff(w(x,y),y) = b*tan(lambda*x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\mu \,a}\ln \left ( \RootOf \left ( \mu \,y-\arctan \left ( 2\,{\frac {{\it \_Z}\,{{\rm e}^{x\mu \,a}}}{{{\it \_Z}}^{2}{{\rm e}^{2\,x\mu \,a}}+1}},-{\frac {{{\it \_Z}}^{2}{{\rm e}^{2\,x\mu \,a}}-1}{{{\it \_Z}}^{2}{{\rm e}^{2\,x\mu \,a}}+1}} \right ) \right ) \right ) } \right ) \left ( 1+ \left ( \tan \left ( \lambda \,x \right ) \right ) ^{2} \right ) ^{1/2\,{\frac {b}{\lambda }}} \]

____________________________________________________________________________________

116.4 Problem 4

problem number 946

Added March 9, 2019.

Problem Chapter 4.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tan (\mu y) w_y = b \sin (\lambda x) w \]

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 = D[w[x, y], x] + a*Tan[mu*y]*D[w[x, y], y] == b*Sin[lambda*x]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to e^{-\frac {b \cos (\lambda x)}{\lambda }} c_1\left (\frac {\log (\sin (\mu y))-a \mu x}{\mu }\right )\right \}\right \} \]

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 :=  diff(w(x,y),x)+ a*tan(mu*y)*diff(w(x,y),y) = b*sin(lambda*x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {\ln \left ( {{\rm e}^{-x\mu \,a}}{\it csgn} \left ( \left ( \cos \left ( \mu \,y \right ) \right ) ^{-1} \right ) \sin \left ( \mu \,y \right ) \right ) }{\mu \,a}} \right ) {{\rm e}^{-{\frac {b\cos \left ( \lambda \,x \right ) }{\lambda }}}} \]

____________________________________________________________________________________

116.5 Problem 5

problem number 947

Added March 9, 2019.

Problem Chapter 4.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \sin (\lambda x) w_x + a w_y = b \cos (\mu y) w \]

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 = Sin[lambda*x]*D[w[x, y], x] + a*D[w[x, y], y] == b*Cos[mu*y]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {b \sin (\mu y)}{a \mu }} c_1\left (\frac {-a \log \left (\sin \left (\frac {\lambda x}{2}\right )\right )+a \log \left (\cos \left (\frac {\lambda x}{2}\right )\right )+\lambda y}{\lambda }\right )\right \}\right \} \]

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 :=  sin(lambda*x)*diff(w(x,y),x)+ a*diff(w(x,y),y) = b*cos(mu*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y\lambda -\ln \left ( \csc \left ( \lambda \,x \right ) -\cot \left ( \lambda \,x \right ) \right ) a}{\lambda }} \right ) {{\rm e}^{{\frac {b\sin \left ( \mu \,y \right ) }{\mu \,a}}}} \]

____________________________________________________________________________________

116.6 Problem 6

problem number 948

Added March 9, 2019.

Problem Chapter 4.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\lambda x) w_x + a w_y = b \tan (\mu y) w \]

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 = Cot[lambda*x]*D[w[x, y], x] + a*D[w[x, y], y] == b*Tan[mu*y]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \cos ^{-\frac {b}{a \mu }}(\mu y) c_1\left (\frac {a \log (\cos (\lambda x))}{\lambda }+y\right )\right \}\right \} \]

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 :=  cot(lambda*x)*diff(w(x,y),x)+ a*diff(w(x,y),y) = b*tan(mu*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {\ln \left ( \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) a-2\,y\lambda -2\,a\ln \left ( \cot \left ( \lambda \,x \right ) \right ) }{\lambda }} \right ) \left ( \cos \left ( \mu \,y \right ) \right ) ^{-{\frac {b}{\mu \,a}}} \]