6.4.17 6.3

6.4.17.1 [1124] Problem 1

problem number 1124

Added March 9, 2019.

Problem Chapter 4.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -x b}{a}\right ) \left (\sec \left (\lambda x +\mu y \right )^{2}\right )^{\frac {c}{2 a \lambda +2 b \mu }}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.17.2 [1125] Problem 2

problem number 1125

Added March 9, 2019.

Problem Chapter 4.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) \left (\sec \left (\lambda x \right )^{2}\right )^{\frac {c}{2 a \lambda }} \left (\sec \left (\mu y \right )^{2}\right )^{\frac {k}{2 \mu b}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.17.3 [1126] Problem 3

problem number 1126

Added March 9, 2019.

Problem Chapter 4.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y}{x}\right ) \left (\sec \left (\lambda x +y \mu \right )^{2}\right )^{\frac {a x}{2 \lambda x +2 y \mu }}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.17.4 [1127] Problem 4

problem number 1127

Added March 9, 2019.

Problem Chapter 4.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b \tan ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda x)\right )}{a \lambda n+a \lambda }\right ) \exp \left (\int _1^x\frac {s \tan ^k\left (\frac {\beta \left (-b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda x)\right ) \tan ^{n+1}(\lambda x)+b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda K[1])\right ) \tan ^{n+1}(\lambda K[1])+a \lambda (n+1) y\right )}{a \lambda (n+1)}\right )+c \tan ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]

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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.17.5 [1128] Problem 5

problem number 1128

Added March 9, 2019.

Problem Chapter 4.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\tan ^{1-n}(\lambda y) \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},-\tan ^2(\lambda y)\right )}{\lambda -\lambda n}-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\tan ^{-n}(\lambda K[1]) \left (s \tan ^k(\beta K[1])+c \tan ^m\left (\frac {-a \mu \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},-\tan ^2(\lambda y)\right ) \tan ^{1-n}(\lambda y)+a \mu \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},-\tan ^2(\lambda K[1])\right ) \tan ^{1-n}(\lambda K[1])+b \lambda \mu x-b \lambda \mu n x}{b \lambda -b \lambda n}\right )\right )}{b}dK[1]\right )\right \}\right \}\]

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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________