6.5.11 4.5
6.5.11.1 [1270] Problem 1
problem number 1270
Added April 4, 2019.
Problem Chapter 5.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \sinh ^k(\lambda x) +c_2 \cosh ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*Sinh[lambda*x]^k+c2*Cosh[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} c_1\left (y-\frac {b x}{a}\right )-\frac {\text {c1} \left (e^{\lambda x}-e^{-\lambda x}\right )^k \left (2-2 e^{2 \lambda x}\right )^{-k} \operatorname {Hypergeometric2F1}\left (-k,-\frac {a k \lambda +1}{2 a \lambda },-\frac {k}{2}-\frac {1}{2 a \lambda }+1,e^{2 \lambda x}\right )}{a k \lambda +1}-\frac {\text {c2} 2^{-n} \left (e^{-\beta y}+e^{\beta y}\right )^n \left (e^{2 \beta y}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {b \beta n+1}{2 b \beta },-\frac {n}{2}-\frac {1}{2 b \beta }+1,-e^{2 \beta y}\right )}{b \beta n+1}\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*sinh(lambda*x)^k+c2*cosh(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\left (\operatorname {c1} \sinh \left (\lambda \textit {\_a} \right )^{k}+\operatorname {c2} \cosh \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n}\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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6.5.11.2 [1271] Problem 2
problem number 1271
Added April 4, 2019.
Problem Chapter 5.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \sinh ^k(\lambda x) \cosh ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ Sinh[lambda*x]^k*Cosh[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cosh ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \sinh ^k(\lambda K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+sinh(lambda*x)^k*cosh(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\sinh \left (\lambda \textit {\_a} \right )^{k} \cosh \left (\frac {\beta \left (a y -b \left (x -\textit {\_a} \right )\right )}{a}\right )^{n} {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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6.5.11.3 [1272] Problem 3
problem number 1272
Added April 4, 2019.
Problem Chapter 5.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + k \tanh (\lambda x)+ s \coth (\mu y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ k*Tanh[lambda*x]+s*coth[mu*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \left (s \coth \left (\mu \left (y+\frac {b (K[1]-x)}{a}\right )\right )+k \tanh (\lambda K[1])\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+k*tanh(lambda*x)+s*coth(mu*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {c \textit {\_a}}{a}} \left (s \coth \left (\frac {\mu \left (a y -b \left (x -\textit {\_a} \right )\right )}{a}\right )+k \tanh \left (\lambda \textit {\_a} \right )\right )d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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6.5.11.4 [1273] Problem 4
problem number 1273
Added April 4, 2019.
Problem Chapter 5.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sinh (\lambda x) w_y = c w + k \cosh (\mu y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Sinh[lambda*x]*D[w[x, y], y] == c*w[x,y]+ k*Cosh[mu*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[2]}{a}} k \cosh \left (\mu \left (y-\int _1^x\frac {b \sinh (\lambda K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \sinh (\lambda K[1])}{a}dK[1]\right )\right )}{a}dK[2]+c_1\left (y-\int _1^x\frac {b \sinh (\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*sinh(lambda*x)*diff(w(x,y),y) = c*w(x,y)+k*cosh(mu*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {k \int _{}^{x}\cosh \left (\frac {\mu \left (y a \lambda -b \cosh \left (\lambda x \right )+b \cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right ) {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {y a \lambda -b \cosh \left (\lambda x \right )}{a \lambda }\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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6.5.11.5 [1274] Problem 5
problem number 1274
Added April 4, 2019.
Problem Chapter 5.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \sinh ^n(\lambda x) w_x + b \cosh ^m(\mu x) w_y = c \cosh ^k(\nu x) w + p \sinh ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Sinh[lambda*x]^n*D[w[x, y], x] + b*Cosh[mu*x]^m*D[w[x, y], y] == c*Cosh[nu*x]^k*w[x,y]+ p*Sinh[beta*y]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cosh ^k(\nu K[2]) \sinh ^{-n}(\lambda K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cosh ^k(\nu K[2]) \sinh ^{-n}(\lambda K[2])}{a}dK[2]\right ) p \sinh ^{-n}(\lambda K[3]) \sinh ^s\left (\beta \left (y-\int _1^x\frac {b \cosh ^m(\mu K[1]) \sinh ^{-n}(\lambda K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \cosh ^m(\mu K[1]) \sinh ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cosh ^m(\mu K[1]) \sinh ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*sinh(lambda*x)^n*diff(w(x,y),x)+ b*cosh(mu*x)^m*diff(w(x,y),y) = c*cosh(nu*x)^k*w(x,y)+p*sinh(beta*y)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {p \int _{}^{x}{\sinh \left (\frac {\beta \left (b \int \cosh \left (\mu \textit {\_f} \right )^{m} \sinh \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f} -b \int \cosh \left (\mu x \right )^{m} \sinh \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{s} \sinh \left (\lambda \textit {\_f} \right )^{-n} {\mathrm e}^{-\frac {c \int \cosh \left (\nu \textit {\_f} \right )^{k} \sinh \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f}}{a}}d \textit {\_f}}{a}+f_{1} \left (-\frac {b \int \cosh \left (\mu x \right )^{m} \sinh \left (\lambda x \right )^{-n}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int \cosh \left (\nu x \right )^{k} \sinh \left (\lambda x \right )^{-n}d x}{a}}\]
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6.5.11.6 [1275] Problem 6
problem number 1275
Added April 4, 2019.
Problem Chapter 5.4.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tanh ^n(\lambda x) w_x + b \coth ^m(\mu x) w_y = c \tanh ^k(\nu y) w + p \coth ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Tanh[lambda*x]^n*D[w[x, y], x] + b*Coth[mu*x]^m*D[w[x, y], y] == c*Tanh[nu*y]^k*w[x,y]+ p*Coth[beta*x]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tanh ^{-n}(\lambda K[2]) \tanh ^k\left (\nu \left (y-\int _1^x\frac {b \coth ^m(\mu K[1]) \tanh ^{-n}(\lambda K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \coth ^m(\mu K[1]) \tanh ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tanh ^{-n}(\lambda K[2]) \tanh ^k\left (\nu \left (y-\int _1^x\frac {b \coth ^m(\mu K[1]) \tanh ^{-n}(\lambda K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \coth ^m(\mu K[1]) \tanh ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) p \coth ^s(\beta K[3]) \tanh ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \coth ^m(\mu K[1]) \tanh ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*tanh(lambda*x)^n*diff(w(x,y),x)+ b*coth(mu*x)^m*diff(w(x,y),y) = c*tanh(nu*y)^k*w(x,y)+p*coth(beta*x)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {p \int _{}^{x}\coth \left (\beta \textit {\_b} \right )^{s} \tanh \left (\lambda \textit {\_b} \right )^{-n} {\mathrm e}^{-\frac {c \int {\tanh \left (\frac {\nu \left (b \int \coth \left (\mu \textit {\_b} \right )^{m} \tanh \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -b \int \coth \left (\mu x \right )^{m} \tanh \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{k} \tanh \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{a}}d \textit {\_b}}{a}+f_{1} \left (-\frac {b \int \coth \left (\mu x \right )^{m} \tanh \left (\lambda x \right )^{-n}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int _{}^{x}{\tanh \left (\frac {\nu \left (b \int \coth \left (\mu \textit {\_b} \right )^{m} \tanh \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -b \int \coth \left (\mu x \right )^{m} \tanh \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{k} \tanh \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{a}}\]
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