6.3.9 4.2

6.3.9.1 [904] Problem 1

problem number 904

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'));
 

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

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6.3.9.2 [905] Problem 2

problem number 905

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]];
 

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

Maple ✓

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

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

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6.3.9.3 [906] Problem 3

problem number 906

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]];
 

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

Maple ✓

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

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

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6.3.9.4 [907] Problem 4

problem number 907

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]];
 

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

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'));
 

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

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6.3.9.5 [908] Problem 5

problem number 908

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]];
 

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

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'));
 

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

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