6.4.8 4.1

6.4.8.1 [1076] Problem 1

problem number 1076

Added Feb. 23, 2019.

Problem Chapter 4.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {1}{2} \left (\frac {c e^{-\lambda x} \left (e^{2 \lambda x}+1\right )}{a \lambda }+\frac {k e^{-\mu y} \left (e^{2 \mu y}+1\right )}{b \mu }\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.8.2 [1077] Problem 2

problem number 1077

Added Feb. 23, 2019.

Problem Chapter 4.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.8.3 [1078] Problem 3

problem number 1078

Added Feb. 23, 2019.

Problem Chapter 4.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) \exp \left (\int _1^xa \sinh \left (\left (\lambda +\frac {\mu y}{x}\right ) K[1]\right )dK[1]\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.8.4 [1079] Problem 4

problem number 1079

Added Feb. 23, 2019.

Problem Chapter 4.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.8.5 [1080] Problem 5

problem number 1080

Added Feb. 23, 2019.

Problem Chapter 4.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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