Added April 3, 2019.
Problem Chapter 5.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \cosh ^k(\lambda x) \cosh ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+Cosh[lambda*x]^k*Cosh[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cosh ^k(\lambda K[1]) \cosh ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+cosh(lambda*x)^k*cosh(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={{\rm e}^{{\frac {cx}{a}}}} \left ( \int ^{x}\!{\frac { \left ( \cosh \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \cosh \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) \]
____________________________________________________________________________________
Added April 3, 2019.
Problem Chapter 5.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c \cosh ^k(\lambda x) w + s \cosh ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Cosh[lambda*x]^k*w[x,y]+ s*Cosh[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c \sqrt {-\sinh ^2(\lambda x)} \text {csch}(\lambda x) \cosh ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cosh ^2(\lambda x)\right )}{a k \lambda +a \lambda }\right ) \left (\int _1^x\frac {\exp \left (\frac {c \cosh ^{k+1}(\lambda K[1]) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cosh ^2(\lambda K[1])\right ) \sinh (\lambda K[1])}{(a \lambda +a k \lambda ) \sqrt {-\sinh ^2(\lambda K[1])}}\right ) s \cosh ^n(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*cosh(lambda*x)^k*w(x,y)+s*cosh(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={{\rm e}^{\int \!{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{k}c}{a}}\,{\rm d}x}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +\int \!{\frac {s \left ( \cosh \left ( \beta \,x \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c\int \! \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x}{a}}}}}\,{\rm d}x \right ) \]
____________________________________________________________________________________
Added April 3, 2019.
Problem Chapter 5.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = \left (c_1 \cosh ^{n_1}(\lambda _1 x)+ c_2 \cosh ^{n_2}(\lambda _2 y) \right ) w + s_1 \cosh ^{k_1}(\beta _1 x)+ s_2 \cosh ^{k_2}(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c1*Cosh[lambda1*x]^n1 + c2*Cosh[lambda2*y]^n2)*w[x,y] + s1*Cosh[beta1*x]^k1+ s2*Cosh[beta2*y]^k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \sqrt {-\sinh ^2(\text {lambda1} x)} \text {csch}(\text {lambda1} x) \cosh ^{\text {n1}+1}(\text {lambda1} x) \, _2F_1\left (\frac {1}{2},\frac {\text {n1}+1}{2};\frac {\text {n1}+3}{2};\cosh ^2(\text {lambda1} x)\right )}{a \text {lambda1} \text {n1}+a \text {lambda1}}+\frac {\text {c2} \sqrt {-\sinh ^2(\text {lambda2} y)} \text {csch}(\text {lambda2} y) \cosh ^{\text {n2}+1}(\text {lambda2} y) \, _2F_1\left (\frac {1}{2},\frac {\text {n2}+1}{2};\frac {\text {n2}+3}{2};\cosh ^2(\text {lambda2} y)\right )}{b \text {lambda2} \text {n2}+b \text {lambda2}}\right ) \left (\int _1^x\frac {\exp \left (\frac {\text {c1} \, _2F_1\left (\frac {1}{2},\frac {\text {n1}+1}{2};\frac {\text {n1}+3}{2};\cosh ^2(\text {lambda1} K[1])\right ) \sinh (\text {lambda1} K[1]) \cosh ^{\text {n1}+1}(\text {lambda1} K[1])}{(a \text {lambda1}+a \text {n1} \text {lambda1}) \sqrt {-\sinh ^2(\text {lambda1} K[1])}}+\frac {\text {c2} \cosh ^{\text {n2}+1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \, _2F_1\left (\frac {1}{2},\frac {\text {n2}+1}{2};\frac {\text {n2}+3}{2};\cosh ^2\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right ) \sinh \left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{(b \text {lambda2}+b \text {n2} \text {lambda2}) \sqrt {-\sinh ^2\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}}\right ) \left (\text {s1} \cosh ^{\text {k1}}(\text {beta1} K[1])+\text {s2} \cosh ^{\text {k2}}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c1*cosh(lambda1*x)^n1 + c2*cosh(lambda2*y)^n2)*w(x,y) + s1*cosh(beta1*x)^k1+ s2*cosh(beta2*y)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \cosh \left ( \lambda 1\,{\it \_a} \right ) \right ) ^{{\it n1}}+{\it c2}\, \left ( \cosh \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \lambda 2}{a}} \right ) \right ) ^{{\it n2}} \right ) }{d{\it \_a}}}} \left ( \int ^{x}\!{\frac {1}{a} \left ( {\it s2}\, \left ( \cosh \left ( {\frac { \left ( ya-b \left ( x-{\it \_b} \right ) \right ) \beta 2}{a}} \right ) \right ) ^{{\it k2}}+{\it s1}\, \left ( \cosh \left ( \beta 1\,{\it \_b} \right ) \right ) ^{{\it k1}} \right ) {{\rm e}^{-{\frac {1}{a}\int \!{\it c1}\, \left ( \cosh \left ( \lambda 1\,{\it \_b} \right ) \right ) ^{{\it n1}}+{\it c2}\, \left ( \cosh \left ( {\frac { \left ( ya-b \left ( x-{\it \_b} \right ) \right ) \lambda 2}{a}} \right ) \right ) ^{{\it n2}}\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) \]
____________________________________________________________________________________
Added April 3, 2019.
Problem Chapter 5.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \cosh (\lambda x+\mu y) w + b \cosh (\nu x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Cosh[lambda*x+my*y]+b*Cosh[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right )+\frac {a x \sinh (\lambda x+\text {my} y)}{\lambda x+\text {my} y}+b \text {Chi}(\nu x)\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*x*cosh(lambda*x+my*y)+b*cosh(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) =-1/2\,{a{{\rm e}^{-\lambda \,x-{\it my}\,y}} \left ( {\frac {{\it my}\,y}{x}}+\lambda \right ) ^{-1}}+1/2\,{a{{\rm e}^{\lambda \,x+{\it my}\,y}} \left ( {\frac {{\it my}\,y}{x}}+\lambda \right ) ^{-1}}-1/2\,b\Ei \left ( 1,\nu \,x \right ) -1/2\,b\Ei \left ( 1,-\nu \,x \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
____________________________________________________________________________________
Added April 3, 2019.
Problem Chapter 5.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \cosh ^n(\lambda x) w_x + b \cosh ^m(\mu x) w_y = c \cosh ^k(\nu x) w + p \cosh ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Cosh[lambda*x]^n*D[w[x, y], x] + b*Cosh[mu*x]^m*D[w[x, y], y] == c*Cosh[nu*x]^k+p*Cosh[beta*y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )+\int _1^x\frac {\cosh ^{-n}(\lambda K[2]) \left (c \cosh ^k(\nu K[2])+p \cosh ^s\left (\beta \left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )\right )\right )}{a}dK[2]\right \}\right \}\]
Maple ✓
restart; pde := a*cosh(lambda*x)^n*diff(w(x,y),x)+ b*cosh(mu*x)^m*diff(w(x,y),y) = c*cosh(nu*x)^k+p*cosh(beta*y)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac { \left ( \cosh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( c \left ( \cosh \left ( \nu \,{\it \_b} \right ) \right ) ^{k}+ \left ( \cosh \left ( {\frac {\beta \, \left ( b\int \! \left ( \cosh \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \cosh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+ya-b\int \! \left ( \cosh \left ( \mu \,x \right ) \right ) ^{m} \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x \right ) }{a}} \right ) \right ) ^{s}p \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {ya-b\int \! \left ( \cosh \left ( \mu \,x \right ) \right ) ^{m} \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x}{a}} \right ) \]
____________________________________________________________________________________
Added April 3, 2019.
Problem Chapter 5.4.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \cosh ^n(\lambda x) w_x + b \cosh ^m(\mu x) w_y = c \cosh ^k(\nu y) w + p \cosh ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Cosh[lambda*x]^n*D[w[x, y], x] + b*Cosh[mu*x]^m*D[w[x, y], y] == c*Cosh[nu*y]^k+p*Cosh[beta*x]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )+\int _1^x\frac {\cosh ^{-n}(\lambda K[2]) \left (c \cosh ^k\left (\nu \left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )\right )+p \cosh ^s(\beta K[2])\right )}{a}dK[2]\right \}\right \}\]
Maple ✓
restart; pde := a*cosh(lambda*x)^n*diff(w(x,y),x)+ b*cosh(mu*x)^m*diff(w(x,y),y) = c*cosh(nu*y)^k+p*cosh(beta*x)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac { \left ( \cosh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \left ( \cosh \left ( {\frac {\nu \, \left ( b\int \! \left ( \cosh \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \cosh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+ya-b\int \! \left ( \cosh \left ( \mu \,x \right ) \right ) ^{m} \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x \right ) }{a}} \right ) \right ) ^{k}c+p \left ( \cosh \left ( \beta \,{\it \_b} \right ) \right ) ^{s} \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {ya-b\int \! \left ( \cosh \left ( \mu \,x \right ) \right ) ^{m} \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x}{a}} \right ) \]
____________________________________________________________________________________