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]];
 

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

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) =   (c*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'));
 

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

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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]];
 

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

Maple ✓

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

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

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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]];
 

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

Maple ✓

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

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

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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]];
 

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

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

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

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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]];
 

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

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'));
 

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

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