#### 6.9.10 4.4

6.9.10.1 [1987] Problem 1
6.9.10.2 [1988] Problem 2
6.9.10.3 [1989] Problem 3
6.9.10.4 [1990] Problem 4
6.9.10.5 [1991] Problem 5

##### 6.9.10.1 [1987] Problem 1

problem number 1987

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]];



$\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \coth ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \coth ^{n+1}(\beta K[1]) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\beta K[1])\right )}{n \beta +\beta }\right ) k \coth ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}$

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'));



$w \left ( x,y,z \right ) ={{\rm e}^{\int \! \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}c\,{\rm d}x}} \left ( {\it \_F1} \left ( -ax+y,-bx+z \right ) +\int \!k \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{m}{{\rm e}^{-c\int \! \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x}}\,{\rm d}x \right )$

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##### 6.9.10.2 [1988] Problem 2

problem number 1988

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'));



$w \left ( x,y,z \right ) ={{\rm e}^{{\frac {qx}{a}}}} \left ( {\rm coth} \left (\lambda \,x\right )-1 \right ) ^{-1/2\,{\frac {p}{a\lambda }}} \left ( {\rm coth} \left (\lambda \,x\right )+1 \right ) ^{-1/2\,{\frac {p}{a\lambda }}} \left ( \int \!{\frac {k{\rm coth} \left (\gamma \,x\right )}{a}{{\rm e}^{-{\frac {qx}{a}}}} \left ( {\rm coth} \left (\lambda \,x\right )-1 \right ) ^{1/2\,{\frac {p}{a\lambda }}} \left ( {\rm coth} \left (\lambda \,x\right )+1 \right ) ^{1/2\,{\frac {p}{a\lambda }}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ay-bx}{a}},-1/2\,{\frac {1}{c\beta } \left ( 2\,xc\beta +\ln \left ( \RootOf \left ( \beta \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a-a\ln \left ( {\frac { \left ( \RootOf \left ( \beta \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \beta \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) \right ) } \right ) \right )$

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##### 6.9.10.3 [1989] Problem 3

problem number 1989

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]];



$\left \{\left \{w(x,y,z)\to e^{c x} c_1\left (z-\frac {b \coth ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\coth ^2(\lambda x)\right )}{k \lambda +\lambda },y-\frac {a \coth ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\beta x)\right )}{\beta n+\beta }\right )-\frac {k \left (e^{-2 \mu x}-1\right )^{-m} \left (e^{-2 \mu x}+1\right )^{-m} \left (-e^{-4 \mu x} \left (e^{2 \mu x}-1\right )^2\right )^m \coth ^m(\mu x) F_1\left (\frac {c}{2 \mu };-m,m;\frac {c}{2 \mu }+1;-e^{-2 \mu x},e^{-2 \mu x}\right )}{c}\right \}\right \}$

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'));



$w \left ( x,y,z \right ) ={{\rm e}^{xc}} \left ( {\it \_F1} \left ( -\int \!a \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{k}\,{\rm d}x+z \right ) +\int \!k \left ( {\rm coth} \left (\mu \,x\right ) \right ) ^{m}{{\rm e}^{-xc}}\,{\rm d}x \right )$

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##### 6.9.10.4 [1990] Problem 4

problem number 1990

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]];



Failed

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'));



$w \left ( x,y,z \right ) ={{\rm e}^{ax}} \left ( \int \!s \left ( {\rm coth} \left (\mu \,x\right ) \right ) ^{m}{{\rm e}^{-ax}}\,{\rm d}x+{\it \_F1} \left ( -\int \!b \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!c \left ( {\frac {\cosh \left ( \lambda \, \left ( b\int \! \left ( {\rm coth} \left (\beta \,{\it \_b}\right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!b \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) }{\sinh \left ( \lambda \, \left ( b\int \! \left ( {\rm coth} \left (\beta \,{\it \_b}\right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!b \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) }} \right ) ^{k}{d{\it \_b}}+z \right ) \right )$

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##### 6.9.10.5 [1991] Problem 5

problem number 1991

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]];



$\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \coth ^{1-\text {n1}}(\text {lambda1} x) \, _2F_1\left (1,\frac {1}{2}-\frac {\text {n1}}{2};\frac {3}{2}-\frac {\text {n1}}{2};\coth ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (c_1\left (y-\int _1^x\frac {\text {b2} \coth ^{-\text {n1}}(\text {lambda1} K[1]) \coth ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \coth ^{-\text {n1}}(\text {lambda1} K[2]) \coth ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )+\int _1^x\frac {\exp \left (\frac {a \coth ^{1-\text {n1}}(\text {lambda1} K[3]) \, _2F_1\left (1,\frac {1}{2}-\frac {\text {n1}}{2};\frac {3}{2}-\frac {\text {n1}}{2};\coth ^2(\text {lambda1} K[3])\right )}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \coth ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \coth ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \coth ^{\text {k3}}(\text {beta3} K[3])\right ) \coth ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]\right )\right \}\right \}$

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'));

`

$w \left ( x,y,z \right ) ={{\rm e}^{\int \!{\frac {a \left ( {\rm coth} \left (\lambda _{1}\,x\right ) \right ) ^{-n_{1}}}{b_{1}}}\,{\rm d}x}} \left ( \int \!{\frac { \left ( c_{1}\, \left ( {\rm coth} \left (\beta _{1}\,x\right ) \right ) ^{k_{1}}+c_{2}\, \left ( {\rm coth} \left (\beta _{2}\,x\right ) \right ) ^{k_{2}}+c_{3}\, \left ( {\rm coth} \left (\beta _{3}\,x\right ) \right ) ^{k_{3}} \right ) \left ( {\rm coth} \left (\lambda _{1}\,x\right ) \right ) ^{-n_{1}}}{b_{1}}{{\rm e}^{-{\frac {a\int \! \left ( {\rm coth} \left (\lambda _{1}\,x\right ) \right ) ^{-n_{1}}\,{\rm d}x}{b_{1}}}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{b_{1}} \left ( yb_{1}-b_{2}\,\int \! \left ( {\frac {\cosh \left ( \lambda _{2}\,x \right ) }{\sinh \left ( \lambda _{2}\,x \right ) }} \right ) ^{n_{2}} \left ( {\frac {\cosh \left ( \lambda _{1}\,x \right ) }{\sinh \left ( \lambda _{1}\,x \right ) }} \right ) ^{-n_{1}}\,{\rm d}x \right ) },{\frac {1}{b_{1}} \left ( -b_{3}\,\int \! \left ( {\frac {\cosh \left ( \lambda _{3}\,x \right ) }{\sinh \left ( \lambda _{3}\,x \right ) }} \right ) ^{n_{3}} \left ( {\frac {\cosh \left ( \lambda _{1}\,x \right ) }{\sinh \left ( \lambda _{1}\,x \right ) }} \right ) ^{-n_{1}}\,{\rm d}x+zb_{1} \right ) } \right ) \right )$

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