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'));
 

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

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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'));
 

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

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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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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