6.8.8 4.2

6.8.8.1 [1810] Problem 1

problem number 1810

Added Oct 10, 2019.

Problem Chapter 8.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a w_y + b w_z = c \cosh ^n(\beta x) w \]

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]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \sqrt {-\sinh ^2(\beta x)} \text {csch}(\beta x) \cosh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]

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); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{c \int \cosh \left (\beta x \right )^{n}d x}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.8.2 [1811] Problem 2

problem number 1811

Added Oct 10, 2019.

Problem Chapter 8.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c \cosh (\lambda x) w_z = \left ( k \cosh (\beta x)+s \cosh (\gamma z) \right ) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, \frac {z a \lambda -c \sinh \left (\lambda x \right )}{a \lambda }\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (k \cosh \left (\beta \textit {\_a} \right )+s \cosh \left (\frac {\gamma \left (z a \lambda -c \sinh \left (\lambda x \right )+c \sinh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )\right )d \textit {\_a}}{a}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.8.3 [1812] Problem 3

problem number 1812

Added Oct 10, 2019.

Problem Chapter 8.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a \cosh ^n(\beta x) w_y + b \cosh ^k(\lambda x) w_z = c \cosh ^m(\gamma x) w \]

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*Cosh[gamma*x]^m *w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (-a \int \sinh \left (\beta x \right )^{n}d x +y , -b \int \sinh \left (\lambda x \right )^{k}d x +z \right ) {\mathrm e}^{c \int \sinh \left (\gamma x \right )^{m}d x}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.8.4 [1813] Problem 4

problem number 1813

Added Oct 10, 2019.

Problem Chapter 8.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \cosh (\beta y) w_y + c \cosh (\lambda x) w_z = k \cosh (\gamma z) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \beta x +2 \arctan \left ({\mathrm e}^{\beta y}\right ) a}{b \beta }, \frac {z a \lambda -c \sinh \left (\lambda x \right )}{a \lambda }\right ) {\mathrm e}^{\frac {k \int _{}^{x}\cosh \left (\frac {\gamma \left (z a \lambda -c \sinh \left (\lambda x \right )+c \sinh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )d \textit {\_a}}{a}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.8.5 [1814] Problem 5

problem number 1814

Added Oct 10, 2019.

Problem Chapter 8.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a_1 \cosh ^{n_1}(\lambda _1 x) w_x + b_1 \cosh ^{m_1}(\beta _1 y) w_y + c_1 \cosh ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cosh ^{n_2}(\lambda _2 x) w_x + b_2 \cosh ^{m_2}(\beta _2 y) w_y + c_2 \cosh ^{k_2}(\gamma _2 z) \right ) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Cosh[lambda1*x]^n1*D[w[x, y,z], x] + b1*Cosh[beta1*y]^m1*D[w[x, y,z], y] +  c1*Cosh[gamma1*x]^k1*D[w[x, y,z], z]== (a2*Cosh[lambda2*x]^n2+b2*Cosh[beta2*y]^m2+c2*Cosh[gamma2*x]^k2) *w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a1*cosh(lambda1*x)^n1*diff(w(x, y,z), x) + b1*cosh(beta1*y)^m1*diff(w(x, y,z), y) +  c1*cosh(gamma1*x)^k1*diff(w(x,y,z),z)= ( a2*cosh(lambda2*x)^n2+b2*cosh(beta2*y)^m2+c2*cosh(gamma2*x)^k2) *w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\int \cosh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \cosh \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y}{\operatorname {b1}}, -\frac {\operatorname {c1} \int \cosh \left (\gamma \operatorname {1} x \right )^{\operatorname {k1}} \cosh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x}{\operatorname {a1}}+z \right ) {\mathrm e}^{\frac {\int _{}^{x}\left (\operatorname {a2} \cosh \left (\lambda \operatorname {2} \textit {\_f} \right )^{\operatorname {n2}}+\operatorname {b2} {\cosh \left (\beta \operatorname {2} \operatorname {RootOf}\left (\int \cosh \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {b1} +\operatorname {a1} \int \cosh \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y -\int \cosh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {b1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\cosh \left (\beta \operatorname {1} \textit {\_a} \right )^{-\operatorname {m1}}d \textit {\_a} \right )\right )}^{\operatorname {m2}}+\operatorname {c2} \cosh \left (\gamma \operatorname {2} \textit {\_f} \right )^{\operatorname {k2}}\right ) \cosh \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f}}{\operatorname {a1}}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________