6.7.17 6.4
6.7.17.1 [1690] Problem 1
problem number 1690
Added June 26, 2019.
Problem Chapter 7.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \cot ^k(\lambda x)+s \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + c*D[w[x,y,z],z]== c*Cot[lambda*x]^k+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)-\frac {c \cot ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)= c*cot(lambda*x)^k+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \cot \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]
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6.7.17.2 [1691] Problem 2
problem number 1691
Added June 26, 2019.
Problem Chapter 7.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \cot (\gamma z) w_z = k \cot (\lambda x)+ s \cot (\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Cot[gamma*z]*D[w[x,y,z],z]== k*Cot[lambda*x]+s*Cot[beta*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log (\sec (\gamma z))}{\gamma }-\frac {c x}{a}\right )+\frac {k \log (\sin (\lambda x))}{a \lambda }+\frac {s \log (\sin (\beta y))}{b \beta }\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*cot(gamma*z)*diff(w(x,y,z),z)= k*cot(lambda*x)+s*cot(beta*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {2 f_{1} \left (\frac {-y a +x b}{b}, \frac {-2 y c \gamma +\ln \left (\csc \left (\gamma z \right )^{2}\right ) b -2 \ln \left (\cot \left (\gamma z \right )\right ) b}{2 c \gamma }\right ) \lambda a b \beta -k \ln \left (\csc \left (\lambda x \right )^{2}\right ) b \beta -s \ln \left (\csc \left (\beta y \right )^{2}\right ) \lambda a}{2 \lambda a b \beta }\]
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6.7.17.3 [1692] Problem 3
problem number 1692
Added June 26, 2019.
Problem Chapter 7.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cot ^n(\beta x) w_y + b \cot ^k(\lambda x) w_z = c \cot ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Cot[beta*x]^n*D[w[x, y,z], y] + b*Cot[lambda*x]^k*D[w[x,y,z],z]== c*Cot[gamma*x]^m+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \cot ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }+z,\frac {a \cot ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\beta x)\right )}{\beta n+\beta }+y\right )-\frac {c \cot ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*cot(beta*x)^n*diff(w(x,y,z),y)+ b*cot(lambda*x)^k*diff(w(x,y,z),z)= c*cot(gamma*x)^m+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \cot \left (\gamma x \right )^{m}d x +s x +f_{1} \left (-a \int \cot \left (\beta x \right )^{n}d x +y , -b \int \cot \left (\lambda x \right )^{k}d x +z \right )\]
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6.7.17.4 [1693] Problem 4
problem number 1693
Added June 26, 2019.
Problem Chapter 7.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cot ^n(\lambda x) w_y + b \cot ^m(\beta y) w_z = c \cot ^k(\gamma y)+s \cot ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Cot[lambda*x]^n*D[w[x, y,z], y] + b*Cot[beta*x]^m*D[w[x,y,z],z]== c*Cot[gamma*y]^k+s*Cot[mu*z]^r;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \cot ^k\left (\frac {\gamma \left (a \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right ) \cot ^{n+1}(\lambda x)+\lambda (n+1) y-a \cot ^{n+1}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda K[1])\right )\right )}{\lambda (n+1)}\right )+s \cot ^r\left (\frac {\mu \left (b \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\beta x)\right ) \cot ^{m+1}(\beta x)+\beta (m+1) z-b \cot ^{m+1}(\beta K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\beta K[1])\right )\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (\frac {b \cot ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\beta x)\right )}{\beta m+\beta }+z,\frac {a \cot ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right )}{\lambda n+\lambda }+y\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*cot(lambda*x)^n*diff(w(x,y,z),y)+ b*cot(beta*x)^m*diff(w(x,y,z),z)= c*cot(gamma*y)^k+s*cot(mu*z)^r;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \cot \left (\lambda x \right )^{n}d x +y , -b \int \cot \left (\beta x \right )^{m}d x +z \right )+\int _{}^{x}\left (s {\left (\frac {1+\cot \left (\mu \left (b \int \cot \left (\beta x \right )^{m}d x -z \right )\right ) \cot \left (\mu b \int \cot \left (\beta \textit {\_f} \right )^{m}d \textit {\_f} \right )}{\cot \left (\mu \left (b \int \cot \left (\beta x \right )^{m}d x -z \right )\right )-\cot \left (\mu b \int \cot \left (\beta \textit {\_f} \right )^{m}d \textit {\_f} \right )}\right )}^{r}+c {\left (\frac {1+\cot \left (\gamma \left (a \int \cot \left (\lambda x \right )^{n}d x -y \right )\right ) \cot \left (\gamma a \int \cot \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} \right )}{\cot \left (\gamma \left (a \int \cot \left (\lambda x \right )^{n}d x -y \right )\right )-\cot \left (\gamma a \int \cot \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} \right )}\right )}^{k}\right )d \textit {\_f}\]
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6.7.17.5 [1694] Problem 5
problem number 1694
Added June 26, 2019.
Problem Chapter 7.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cot (\beta x) w_y + c \cot (\lambda x) w_z = k \cot (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*Cot[beta*x]*D[w[x, y,z], y] + c*Cot[lambda*x]*D[w[x,y,z],z]== k*Cot[gamma*z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \cot \left (\frac {\gamma (a \lambda z-c \log (\sin (\lambda x))+c \log (\sin (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b \log (\sin (\beta x))}{a \beta },z-\frac {c \log (\sin (\lambda x))}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*cot(beta*x)*diff(w(x,y,z),y)+ c*cot(lambda*x)*diff(w(x,y,z),z)= k*cot(gamma*z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {k \int _{}^{x}\frac {1+\cot \left (\frac {\gamma c \ln \left (\csc \left (\lambda \textit {\_a} \right )^{2}\right )}{2 a \lambda }\right ) \cot \left (\frac {\gamma \left (2 z a \lambda +c \ln \left (\csc \left (\lambda x \right )^{2}\right )\right )}{2 a \lambda }\right )}{\cot \left (\frac {\gamma c \ln \left (\csc \left (\lambda \textit {\_a} \right )^{2}\right )}{2 a \lambda }\right )-\cot \left (\frac {\gamma \left (2 z a \lambda +c \ln \left (\csc \left (\lambda x \right )^{2}\right )\right )}{2 a \lambda }\right )}d \textit {\_a}}{a}+f_{1} \left (\frac {2 y a \beta +b \ln \left (\csc \left (\beta x \right )^{2}\right )}{2 \beta a}, \frac {2 z a \lambda +c \ln \left (\csc \left (\lambda x \right )^{2}\right )}{2 a \lambda }\right )\]
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