6.9.10 4.4
6.9.10.1 [1986] Problem 1
problem number 1986
Added Jan 19, 2020.
Problem Chapter 9.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \coth ^n(\beta x) w + k \coth ^m(\lambda x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+ a*D[w[x,y,z],y]+b*D[w[x,y,z],z]==c*Coth[beta*x]^n*w[x,y,z]+ k*Coth[lambda*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \coth ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\coth ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \coth ^{n+1}(\beta K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\coth ^2(\beta K[1])\right )}{n \beta +\beta }\right ) k \coth ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
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*coth(beta*x)^n*w(x,y,z)+ k*coth(lambda*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (k \int \coth \left (\lambda x \right )^{m} {\mathrm e}^{-c \int \coth \left (\beta x \right )^{n}d x}d x +f_{1} \left (-a x +y , -b x +z \right )\right ) {\mathrm e}^{c \int \coth \left (\beta x \right )^{n}d x}\]
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6.9.10.2 [1987] Problem 2
problem number 1987
Added Jan 19, 2020.
Problem Chapter 9.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \coth (\beta z) w_z = \left (p \coth (\lambda x) + q \right ) w + k \coth (\gamma x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x,y,z],x]+ b*D[w[x,y,z],y]+c*Coth[beta*z]*D[w[x,y,z],z]==(p*Coth[lambda*x]+q)*w[x,y,z]+ k*Coth[gamma*x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\begin{align*}& \left \{w(x,y,z)\to e^{\frac {q x}{a}} \sinh ^{\frac {p}{a \lambda }}(\lambda x) \left (\int _1^x\frac {e^{-\frac {q K[1]}{a}} k \coth (\gamma K[1]) \sinh ^{-\frac {p}{a \lambda }}(\lambda K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a},\frac {\log (\cosh (\beta z))}{\beta }-\frac {c x}{a}\right )\right )\right \}\\& \left \{w(x,y,z)\to e^{\frac {q x}{a}} \sinh ^{\frac {p}{a \lambda }}(\lambda x) \left (\int _1^x\frac {e^{-\frac {q K[2]}{a}} k \coth (\gamma K[2]) \sinh ^{-\frac {p}{a \lambda }}(\lambda K[2])}{a}dK[2]+c_1\left (y-\frac {b x}{a},\frac {\log (\cosh (\beta z))}{\beta }-\frac {c x}{a}\right )\right )\right \}\\\end{align*}
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*coth(beta*z)*diff(w(x,y,z),z)=(p*coth(lambda*x)+q)*w(x,y,z)+ k*coth(gamma*x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = {\mathrm e}^{\frac {y q}{b}} \sinh \left (\lambda x \right )^{\frac {p}{\lambda a}} \left (\frac {k \int _{}^{y}{\mathrm e}^{-\frac {\textit {\_a} q}{b}} \coth \left (\gamma \left (\frac {\textit {\_a} a}{b}-\frac {a y}{b}+x \right )\right ) \left (-\sinh \left (\frac {\left (\left (-\textit {\_a} +y \right ) a -b x \right ) \lambda }{b}\right )\right )^{-\frac {p}{\lambda a}}d \textit {\_a}}{b}+f_{1} \left (-\frac {a y}{b}+x , -y +\frac {b \ln \left (\frac {\coth \left (\beta z \right )^{2}}{\coth \left (\beta z \right )-1}\right )}{2 c \beta }-\frac {b \ln \left (\coth \left (\beta z \right )+1\right )}{2 c \beta }\right )\right )\]
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6.9.10.3 [1988] Problem 3
problem number 1988
Added Jan 19, 2020.
Problem Chapter 9.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \coth ^n(\beta x) w_y + b \coth ^k(\lambda x) w_z = c w + s \coth ^m(\mu x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+ a*Coth[beta*x]^n*D[w[x,y,z],y]+b*Coth[lambda*x]^k*D[w[x,y,z],z]==c*w[x,y,z]+ k*Coth[mu*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{c x} \left (\int _1^xe^{-c K[1]} k \coth ^m(\mu K[1])dK[1]+c_1\left (z-\frac {b \coth ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\lambda x)\right )}{k \lambda +\lambda },y-\frac {a \coth ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\coth ^2(\beta x)\right )}{\beta n+\beta }\right )\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*coth(beta*x)^n*diff(w(x,y,z),y)+ b*coth(lambda*x)^k*diff(w(x,y,z),z)=c*w(x,y,z)+ k*coth(mu*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (k \int \coth \left (\mu x \right )^{m} {\mathrm e}^{-c x}d x +f_{1} \left (-a \int \coth \left (\beta x \right )^{n}d x +y , -b \int \coth \left (\lambda x \right )^{k}d x +z \right )\right ) {\mathrm e}^{c x}\]
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6.9.10.4 [1989] Problem 4
problem number 1989
Added Jan 19, 2020.
Problem Chapter 9.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + b \coth ^n(\beta x) w_y + c \coth ^k(\lambda y) w_z = a w + s \coth ^m(\mu x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+ b*Coth[beta*x]^n*D[w[x,y,z],y]+c*Coth[lambda*y]^k*D[w[x,y,z],z]==a*w[x,y,z]+ s*Coth[mu*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{a x} \left (\int _1^xe^{-a K[2]} s \coth ^m(\mu K[2])dK[2]+c_1\left (y-\frac {b \coth ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\coth ^2(\beta x)\right )}{\beta n+\beta },z-\int _1^xc \coth ^k\left (\frac {\lambda \left (-b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\coth ^2(\beta x)\right ) \coth ^{n+1}(\beta x)+\beta (n+1) y+b \coth ^{n+1}(\beta K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\coth ^2(\beta K[1])\right )\right )}{\beta (n+1)}\right )dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ b*coth(beta*x)^n*diff(w(x,y,z),y)+ c*coth(lambda*y)^k*diff(w(x,y,z),z)=a*w(x,y,z)+ s*coth(mu*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (s \int \coth \left (\mu x \right )^{m} {\mathrm e}^{-a x}d x +f_{1} \left (-b \int \coth \left (\beta x \right )^{n}d x +y , -c \int _{}^{x}{\coth \left (\lambda \left (b \int \coth \left (\beta \textit {\_b} \right )^{n}d \textit {\_b} -b \int \coth \left (\beta x \right )^{n}d x +y \right )\right )}^{k}d \textit {\_b} +z \right )\right ) {\mathrm e}^{a x}\]
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6.9.10.5 [1990] Problem 5
problem number 1990
Added Jan 19, 2020.
Problem Chapter 9.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b_1 \coth ^{n_1}(\lambda _1 x) w_x + b_2 \coth ^{n_2}(\lambda _2 y) w_y + b_3 \coth ^{n_3}(\lambda _3 z) w_z = a w + c_1 \coth ^{k_1}(\beta _1 x)+ c_2 \coth ^{k_2}(\beta _2 y)+ c_3 \coth ^{k_3}(\beta _3 z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = b1*Coth[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Coth[lambda2*x]^n2*D[w[x,y,z],y]+b3*Coth[lambda3*x]^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Coth[beta1*x]^k1+ c2*Coth[beta2*x]^k2+ c3*Coth[beta3*x]^k3;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \coth ^{1-\text {n1}}(\text {lambda1} x) \operatorname {Hypergeometric2F1}\left (1,\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},\coth ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (\int _1^x\frac {\exp \left (\frac {a \coth ^{1-\text {n1}}(\text {lambda1} K[3]) \operatorname {Hypergeometric2F1}\left (1,\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},\coth ^2(\text {lambda1} K[3])\right )}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \coth ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \coth ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \coth ^{\text {k3}}(\text {beta3} K[3])\right ) \coth ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b2} \coth ^{-\text {n1}}(\text {lambda1} K[1]) \coth ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \coth ^{-\text {n1}}(\text {lambda1} K[2]) \coth ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := b__1*coth(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*coth(lambda__2*x)^(n__2)*diff(w(x,y,z),y)+ b__3*coth(lambda__3*x)^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*coth(beta__1*x)^(k__1)+ c__2*coth(beta__2*x)^(k__2)+ c__3*coth(beta__3*x)^(k__3);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\frac {\int {\mathrm e}^{-\frac {a \int \coth \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}} \left (c_{1} \coth \left (\beta _{1} x \right )^{k_{1}}+c_{2} \coth \left (\beta _{2} x \right )^{k_{2}}+c_{3} \coth \left (\beta _{3} x \right )^{k_{3}}\right ) \coth \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}+f_{1} \left (-\frac {b_{2} \int \coth \left (\lambda _{2} x \right )^{n_{2}} \coth \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}+y , -\frac {b_{3} \int \coth \left (\lambda _{3} x \right )^{n_{3}} \coth \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}+z \right )\right ) {\mathrm e}^{\frac {a \int \coth \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}}\]
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