7.5.8 4.2

7.5.8.1 [1256] Problem 1

problem number 1256

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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\cosh ^{k}\left (\textit {\_a} \lambda \right )\right ) \left (\cosh ^{n}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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7.5.8.2 [1257] Problem 2

problem number 1257

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'));
 

\[w \left (x , y\right ) = \left (\int \frac {s \left (\cosh ^{n}\left (\beta x \right )\right ) {\mathrm e}^{-\frac {c \left (\int \left (\cosh ^{k}\left (\lambda x \right )\right )d x \right )}{a}}}{a}d x +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\cosh ^{k}\left (\lambda x \right )\right )}{a}d x}\]

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7.5.8.3 [1258] Problem 3

problem number 1258

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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {s1} \left (\cosh ^{\mathit {k1}}\left (\textit {\_b} \beta 1 \right )\right )+\mathit {s2} \left (\cosh ^{\mathit {k2}}\left (\frac {\left (a y -\left (-\textit {\_b} +x \right ) b \right ) \beta 2}{a}\right )\right )\right ) {\mathrm e}^{-\frac {\int \left (\mathit {c1} \left (\cosh ^{\mathit {n1}}\left (\textit {\_b} \lambda 1 \right )\right )+\mathit {c2} \left (\cosh ^{\mathit {n2}}\left (\frac {\left (a y -\left (-\textit {\_b} +x \right ) b \right ) \lambda 2}{a}\right )\right )\right )d \textit {\_b}}{a}}}{a}d \textit {\_b} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {c1} \left (\cosh ^{\mathit {n1}}\left (\textit {\_a} \lambda 1 \right )\right )+\mathit {c2} \left (\cosh ^{\mathit {n2}}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )\right )}{a}d \textit {\_a}}\]

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7.5.8.4 [1259] Problem 4

problem number 1259

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'));
 

\[w \left (x , y\right ) = \frac {a \sinh \left (\lambda x +\mathit {my} y \right )}{\lambda +\frac {\mathit {my} y}{x}}+b \Chi \left (\nu x \right )+\textit {\_F1} \left (\frac {y}{x}\right )\]

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7.5.8.5 [1260] Problem 5

problem number 1260

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]];
 

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

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'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {\left (c \left (\cosh ^{k}\left (\textit {\_b} \nu \right )\right )+p \left (\cosh ^{s}\left (\frac {\left (a y +b \left (\int \left (\cosh ^{-n}\left (\textit {\_b} \lambda \right )\right ) \left (\cosh ^{m}\left (\textit {\_b} \mu \right )\right )d \textit {\_b} \right )-b \left (\int \left (\cosh ^{-n}\left (\lambda x \right )\right ) \left (\cosh ^{m}\left (\mu x \right )\right )d x \right )\right ) \beta }{a}\right )\right )\right ) \left (\cosh ^{-n}\left (\textit {\_b} \lambda \right )\right )}{a}d \textit {\_b} +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\cosh ^{-n}\left (\lambda x \right )\right ) \left (\cosh ^{m}\left (\mu x \right )\right )d x \right )}{a}\right )\]

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7.5.8.6 [1261] Problem 6

problem number 1261

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]];
 

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

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'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {\left (c \left (\cosh ^{k}\left (\frac {\left (a y +b \left (\int \left (\cosh ^{-n}\left (\textit {\_b} \lambda \right )\right ) \left (\cosh ^{m}\left (\textit {\_b} \mu \right )\right )d \textit {\_b} \right )-b \left (\int \left (\cosh ^{-n}\left (\lambda x \right )\right ) \left (\cosh ^{m}\left (\mu x \right )\right )d x \right )\right ) \nu }{a}\right )\right )+p \left (\cosh ^{s}\left (\textit {\_b} \beta \right )\right )\right ) \left (\cosh ^{-n}\left (\textit {\_b} \lambda \right )\right )}{a}d \textit {\_b} +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\cosh ^{-n}\left (\lambda x \right )\right ) \left (\cosh ^{m}\left (\mu x \right )\right )d x \right )}{a}\right )\]

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