#### 6.9.8 4.2

6.9.8.1 [1977] Problem 1
6.9.8.2 [1978] Problem 2
6.9.8.3 [1979] Problem 3
6.9.8.4 [1980] Problem 4
6.9.8.5 [1981] Problem 5

##### 6.9.8.1 [1977] Problem 1

problem number 1977

Problem Chapter 9.4.2.1, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

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



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

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



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

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##### 6.9.8.2 [1978] Problem 2

problem number 1978

Problem Chapter 9.4.2.2, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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



$\left \{\left \{w(x,y,z)\to e^{\frac {p \sinh (\lambda x)+\lambda q x}{a \lambda }} \left (\int _1^x\frac {e^{-\frac {\lambda q K[1]+p \sinh (\lambda K[1])}{a \lambda }} k \cosh (\gamma K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a},\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\beta z}{2}\right )\right )}{\beta }-\frac {c x}{a}\right )\right )\right \}\right \}$

Maple

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



$w \left ( x,y,z \right ) ={{\rm e}^{{\frac {qx\lambda +p\sinh \left ( \lambda \,x \right ) }{a\lambda }}}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {-xc\beta +2\,\arctan \left ( {{\rm e}^{\beta \,z}} \right ) a}{c\beta }} \right ) +\int \!{\frac {k\cosh \left ( \gamma \,x \right ) }{a}{{\rm e}^{{\frac {-qx\lambda -p\sinh \left ( \lambda \,x \right ) }{a\lambda }}}}}\,{\rm d}x \right )$

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##### 6.9.8.3 [1979] Problem 3

problem number 1979

Problem Chapter 9.4.2.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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



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

Maple

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



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

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##### 6.9.8.4 [1980] Problem 4

problem number 1980

Problem Chapter 9.4.2.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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



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

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##### 6.9.8.5 [1981] Problem 5

problem number 1981

Problem Chapter 9.4.2.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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



$\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \sqrt {-\sinh ^2(\text {lambda1} x)} \text {csch}(\text {lambda1} x) \cosh ^{1-\text {n1}}(\text {lambda1} x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},\cosh ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (c_1\left (y-\int _1^x\frac {\text {b2} \cosh ^{-\text {n1}}(\text {lambda1} K[1]) \cosh ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \cosh ^{-\text {n1}}(\text {lambda1} K[2]) \cosh ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )+\int _1^x\frac {\exp \left (\frac {a \cosh ^{1-\text {n1}}(\text {lambda1} K[3]) \text {csch}(\text {lambda1} K[3]) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},\cosh ^2(\text {lambda1} K[3])\right ) \sqrt {-\sinh ^2(\text {lambda1} K[3])}}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \cosh ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \cosh ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \cosh ^{\text {k3}}(\text {beta3} K[3])\right ) \cosh ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]\right )\right \}\right \}$

Maple

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

`

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

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