6.4.9 4.2

6.4.9.1 [1082] Problem 1

problem number 1082

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'));
 

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

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6.4.9.2 [1083] Problem 2

problem number 1083

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'));
 

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

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6.4.9.3 [1084] Problem 3

problem number 1084

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) {{\rm e}^{{a\sinh \left ( \lambda \,x+\mu \,y \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}}}\]

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6.4.9.4 [1085] Problem 4

problem number 1085

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'));
 

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

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6.4.9.5 [1086] Problem 5

problem number 1086

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'));
 

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

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