6.3.18 6.3

6.3.18.1 [946] Problem 1

problem number 946

Added Feb. 11, 2019.

Problem Chapter 3.6.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b y^n w_y = c \tan (\lambda x) + k \tan (\mu y) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{2\,a\lambda \,\mu \,b} \left ( 2\,{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \mu \,ba\lambda +c\ln \left ( 1+ \left ( \tan \left ( x\lambda \right ) \right ) ^{2} \right ) \mu \,b+k\ln \left ( 1+ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2} \right ) a\lambda \right ) }\]

____________________________________________________________________________________

6.3.18.2 [947] Problem 2

problem number 947

Added Feb. 11, 2019.

Problem Chapter 3.6.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b y^n w_y = c \tan (\lambda x+\mu y) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {c\ln \left ( 1+ \left ( \tan \left ( x\lambda +\mu \,y \right ) \right ) ^{2} \right ) }{2\,a\lambda +2\,\mu \,b}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \]

____________________________________________________________________________________

6.3.18.3 [948] Problem 3

problem number 948

Added Feb. 11, 2019.

Problem Chapter 3.6.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

____________________________________________________________________________________

6.3.18.4 [949] Problem 4

problem number 949

Added Feb. 11, 2019.

Problem Chapter 3.6.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \tan ^k\left (\frac {\beta \left (-b \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\lambda x)\right ) \tan ^{n+1}(\lambda x)+b \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\lambda K[1])\right ) \tan ^{n+1}(\lambda K[1])+a \lambda (n+1) y\right )}{a \lambda (n+1)}\right )+c \tan ^m(\mu K[1])}{a}dK[1]+c_1\left (y-\frac {b \tan ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\lambda x)\right )}{a \lambda n+a \lambda }\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( \tan \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( { \left ( \tan \left ( \beta \, \left ( -\int \!{\frac {b \left ( \tan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \right ) +\tan \left ( {\frac {\beta \,b\int \! \left ( \tan \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}}{a}} \right ) \right ) \left ( 1-\tan \left ( \beta \, \left ( -\int \!{\frac {b \left ( \tan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \right ) \tan \left ( {\frac {\beta \,b\int \! \left ( \tan \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}}{a}} \right ) \right ) ^{-1}} \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \tan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \]

____________________________________________________________________________________

6.3.18.5 [950] Problem 5

problem number 950

Added Feb. 11, 2019.

Problem Chapter 3.6.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^y\frac {\tan ^{-n}(\lambda K[1]) \left (s \tan ^k(\beta K[1])+c \tan ^m\left (\frac {-a \mu \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\tan ^2(\lambda y)\right ) \tan ^{1-n}(\lambda y)+a \mu \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\tan ^2(\lambda K[1])\right ) \tan ^{1-n}(\lambda K[1])+b \lambda \mu x-b \lambda \mu n x}{b \lambda -b \lambda n}\right )\right )}{b}dK[1]+c_1\left (\frac {\tan ^{1-n}(\lambda y) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\tan ^2(\lambda y)\right )}{\lambda -\lambda n}-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {a\int \! \left ( \tan \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) +\int ^{y}\!{\frac { \left ( \tan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( s \left ( \tan \left ( \beta \,{\it \_b} \right ) \right ) ^{k}+ \left ( { \left ( -\tan \left ( \mu \, \left ( -{\frac {a\int \! \left ( \tan \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) -\tan \left ( {\frac {\mu \,a\int \! \left ( \tan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}}{b}} \right ) \right ) \left ( \tan \left ( \mu \, \left ( -{\frac {a\int \! \left ( \tan \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \tan \left ( {\frac {\mu \,a\int \! \left ( \tan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}}{b}} \right ) -1 \right ) ^{-1}} \right ) ^{m}c \right ) }{d{\it \_b}}\]

____________________________________________________________________________________