6.4.9 4.2

6.4.9.1 [1081] Problem 1

problem number 1081

Added Feb. 23, 2019.

Problem Chapter 4.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c \cosh (\lambda x) + k \cosh (\mu y)) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.9.2 [1082] Problem 2

problem number 1082

Added Feb. 23, 2019.

Problem Chapter 4.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c \cosh (\lambda x +\mu y) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.9.3 [1083] Problem 3

problem number 1083

Added Feb. 23, 2019.

Problem Chapter 4.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \cosh (\lambda x +\mu y) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {y}{x}\right ) {\mathrm e}^{\frac {a \sinh \left (\lambda x +\mu y \right )}{\lambda +\frac {\mu y}{x}}}\]

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6.4.9.4 [1084] Problem 4

problem number 1084

Added Feb. 23, 2019.

Problem Chapter 4.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

$Aborted

Maple ✓

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

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

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6.4.9.5 [1085] Problem 5

problem number 1085

Added Feb. 23, 2019.

Problem Chapter 4.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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