#### 6.9.9 4.3

6.9.9.1 [1982] Problem 1
6.9.9.2 [1983] Problem 2
6.9.9.3 [1984] Problem 3
6.9.9.4 [1985] Problem 4
6.9.9.5 [1986] Problem 5

##### 6.9.9.1 [1982] Problem 1

problem number 1982

Problem Chapter 9.4.3.1, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + a w_y + b w_z = c \tanh ^n(\beta x) w + k \tanh ^m(\lambda x)$

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*Tanh[beta*x]^n*w[x,y,z]+ k*Tanh[lambda*x]^m;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{n+1}(\beta x) \text {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \text {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta K[1])\right ) \tanh ^{n+1}(\beta K[1])}{n \beta +\beta }\right ) k \tanh ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}$

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*tanh(beta*x)^n*w(x,y,z)+ k*tanh(lambda*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



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

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##### 6.9.9.2 [1983] Problem 2

problem number 1983

Problem Chapter 9.4.3.2, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $a w_x + b w_y + c \tanh (\beta z) w_z = \left (p \tanh (\lambda x) + q \right ) w + k \tanh (\gamma x)$

Mathematica

ClearAll["Global*"];
pde =  a*D[w[x,y,z],x]+ b*D[w[x,y,z],y]+c*Tanh[beta*z]*D[w[x,y,z],z]==(p*Tanh[lambda*x]+q)*w[x,y,z]+ k*Tanh[gamma*x];
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to e^{\frac {q x}{a}} \cosh ^{\frac {p}{a \lambda }}(\lambda x) \left (\int _1^x\frac {e^{-\frac {q K[1]}{a}} k \cosh ^{-\frac {p}{a \lambda }}(\lambda K[1]) \tanh (\gamma K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a},\frac {\log (\sinh (\beta z))}{\beta }-\frac {c x}{a}\right )\right )\right \}\right \}$

Maple

restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*tanh(beta*z)*diff(w(x,y,z),z)=(p*tanh(lambda*x)+q)*w(x,y,z)+ k*tanh(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 ) = \left ( -\tanh \left ( \lambda \,x \right ) +1 \right ) ^{-1/2\,{\frac {q+p}{a\lambda }}} \left ( -\tanh \left ( \lambda \,x \right ) -1 \right ) ^{-1/2\,{\frac {-q+p}{a\lambda }}} \left ( -\int ^{z}\!{\frac {k}{\tanh \left ( \beta \,{\it \_a} \right ) c}\tanh \left ( 1/2\,{\frac {\gamma \, \left ( -2\,xc\beta -a\ln \left ( \tanh \left ( \beta \,z \right ) -1 \right ) -a\ln \left ( \tanh \left ( \beta \,z \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,z \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) \right ) }{c\beta }} \right ) \left ( \tanh \left ( 1/2\,{\frac {\lambda \, \left ( -2\,xc\beta -a\ln \left ( \tanh \left ( \beta \,z \right ) -1 \right ) -a\ln \left ( \tanh \left ( \beta \,z \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,z \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) \right ) }{c\beta }} \right ) +1 \right ) ^{1/2\,{\frac {q+p}{a\lambda }}} \left ( \tanh \left ( 1/2\,{\frac {\lambda \, \left ( -2\,xc\beta -a\ln \left ( \tanh \left ( \beta \,z \right ) -1 \right ) -a\ln \left ( \tanh \left ( \beta \,z \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,z \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) \right ) }{c\beta }} \right ) -1 \right ) ^{1/2\,{\frac {-q+p}{a\lambda }}}}{d{\it \_a}}+{\it \_F1} \left ( 1/2\,{\frac {2\,xc\beta +a\ln \left ( \tanh \left ( \beta \,z \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,z \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,z \right ) \right ) }{c\beta }},1/2\,{\frac {2\,yc\beta +b\ln \left ( \tanh \left ( \beta \,z \right ) -1 \right ) +b\ln \left ( \tanh \left ( \beta \,z \right ) +1 \right ) -2\,b\ln \left ( \tanh \left ( \beta \,z \right ) \right ) }{c\beta }} \right ) \right )$

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##### 6.9.9.3 [1984] Problem 3

problem number 1984

Problem Chapter 9.4.3.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + a \tanh ^n(\beta x) w_y + b \tanh ^k(\lambda x) w_z = c w + s \tanh ^m(\mu x)$

Mathematica

ClearAll["Global*"];
pde =  D[w[x,y,z],x]+ a*Tanh[beta*x]^n*D[w[x,y,z],y]+b*Tanh[lambda*x]^k*D[w[x,y,z],z]==c*w[x,y,z]+ k*Tanh[mu*x]^m;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



$\left \{\left \{w(x,y,z)\to e^{c x} c_1\left (y-\frac {a \tanh ^{n+1}(\beta x) \text {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tanh ^{k+1}(\lambda x) \text {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )-\frac {k \left (e^{-2 \mu x}-1\right )^m \left (e^{-2 \mu x}+1\right )^m \left (-e^{-4 \mu x} \left (e^{2 \mu x}-1\right )^2\right )^{-m} \tanh ^m(\mu x) F_1\left (\frac {c}{2 \mu };m,-m;\frac {c}{2 \mu }+1;-e^{-2 \mu x},e^{-2 \mu x}\right )}{c}\right \}\right \}$

Maple

restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*tanh(beta*x)^n*diff(w(x,y,z),y)+ b*tanh(lambda*x)^k*diff(w(x,y,z),z)=c*w(x,y,z)+ k*tanh(mu*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



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

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##### 6.9.9.4 [1985] Problem 4

problem number 1985

Problem Chapter 9.4.3.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $w_x + b \tanh ^n(\beta x) w_y + c \tanh ^k(\lambda y) w_z = a w + s \tanh ^m(\mu x)$

Mathematica

ClearAll["Global*"];
pde =  D[w[x,y,z],x]+ b*Tanh[beta*x]^n*D[w[x,y,z],y]+c*Tanh[lambda*y]^k*D[w[x,y,z],z]==a*w[x,y,z]+ s*Tanh[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*tanh(beta*x)^n*diff(w(x,y,z),y)+ c*tanh(lambda*y)^k*diff(w(x,y,z),z)=a*w(x,y,z)+ s*tanh(mu*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));



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

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##### 6.9.9.5 [1986] Problem 5

problem number 1986

Problem Chapter 9.4.3.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y,z)$$ $b_1 \tanh ^{n_1}(\lambda _1 x) w_x + b_2 \tanh ^{n_2}(\lambda _2 y) w_y + b_3 \tanh ^{n_3}(\lambda _3 z) w_z = a w + c_1 \tanh ^{k_1}(\beta _1 x)+ c_2 \tanh ^{k_2}(\beta _2 y)+ c_3 \tanh ^{k_3}(\beta _3 z)$

Mathematica

ClearAll["Global*"];
pde =  b1*Tanh[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Tanh[lambda2*x]^n2*D[w[x,y,z],y]+b3*Tanh[lambda3*x]^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Tanh[beta1*x]^k1+ c2*Tanh[beta2*x]^k2+ c3*Tanh[beta3*x]^k3;
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];



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

Maple

restart;
local gamma;
pde := b__1*tanh(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*tanh(lambda__2*x)^(n__2)*diff(w(x,y,z),y)+ b__3*tanh(lambda__3*x)^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*tanh(beta__1*x)^(k__1)+ c__2*tanh(beta__2*x)^(k__2)+ c__3*tanh(beta__3*x)^(k__3);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));

`

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

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