6.3.9 4.2

6.3.9.1 [905] Problem 1

problem number 905

Added Feb. 9, 2019.

Problem Chapter 3.4.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{b\mu \,a\lambda } \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +k\sinh \left ( \mu \,y \right ) a\lambda +c\sinh \left ( \lambda \,x \right ) b\mu \right ) }\]

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6.3.9.2 [906] Problem 2

problem number 906

Added Feb. 9, 2019.

Problem Chapter 3.4.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {c\sinh \left ( \lambda \,x+\mu \,y \right ) }{a\lambda +b\mu }}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

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6.3.9.3 [907] Problem 3

problem number 907

Added Feb. 9, 2019.

Problem Chapter 3.4.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{ \left ( a\lambda +b\mu \right ) ^{2}} \left ( \left ( a\lambda +b\mu \right ) ^{2}{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) + \left ( \left ( a\lambda +b\mu \right ) x\sinh \left ( \lambda \,x+\mu \,y \right ) -\cosh \left ( \lambda \,x+\mu \,y \right ) a \right ) a \right ) }\]

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6.3.9.4 [908] Problem 4

problem number 908

Added Feb. 9, 2019.

Problem Chapter 3.4.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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6.3.9.5 [909] Problem 5

problem number 909

Added Feb. 9, 2019.

Problem Chapter 3.4.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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