6.7.8 4.2

6.7.8.1 [1639] Problem 1

problem number 1639

Added June 19, 2019.

Problem Chapter 7.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y + b w_z = c \cosh ^k(\lambda x)+s \]

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

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \sqrt {-\sinh ^2(\lambda x)} \text {csch}(\lambda x) \cosh ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cosh ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]

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

\[w \left ( x,y,z \right ) =\int \!c \left ( \cosh \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]

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6.7.8.2 [1640] Problem 2

problem number 1640

Added June 19, 2019.

Problem Chapter 7.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {za\lambda -c\sinh \left ( x\lambda \right ) }{a\lambda }} \right ) +\int ^{x}\!{\frac {1}{a} \left ( k\cosh \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) +s\cosh \left ( {\frac {\gamma \, \left ( za\lambda -c\sinh \left ( x\lambda \right ) +c\sinh \left ( {\it \_a}\,\lambda \right ) \right ) }{a\lambda }} \right ) \right ) }{d{\it \_a}}\]

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6.7.8.3 [1641] Problem 3

problem number 1641

Added June 19, 2019.

Problem Chapter 7.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \sinh (\beta x) \cosh ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cosh ^2(\beta x)\right )}{(\beta n+\beta ) \sqrt {-\sinh ^2(\beta x)}}+y,\frac {b \sinh (\lambda x) \cosh ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cosh ^2(\lambda x)\right )}{(k \lambda +\lambda ) \sqrt {-\sinh ^2(\lambda x)}}+z\right )+\frac {c \sqrt {-\sinh ^2(\gamma x)} \text {csch}(\gamma x) \cosh ^{m+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cosh ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]

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

\[w \left ( x,y,z \right ) =\int \!c \left ( \cosh \left ( x\gamma \right ) \right ) ^{m}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( \cosh \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cosh \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]

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6.7.8.4 [1642] Problem 4

problem number 1642

Added June 19, 2019.

Problem Chapter 7.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-b\beta \,x+2\,\arctan \left ( {{\rm e}^{\beta \,y}} \right ) a}{\beta \,b}},{\frac {za\lambda -c\sinh \left ( x\lambda \right ) }{a\lambda }} \right ) +\int ^{x}\!{\frac {k}{a}\cosh \left ( {\frac {\gamma \, \left ( za\lambda -c\sinh \left ( x\lambda \right ) +c\sinh \left ( {\it \_a}\,\lambda \right ) \right ) }{a\lambda }} \right ) }{d{\it \_a}}\]

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6.7.8.5 [1643] Problem 5

problem number 1643

Added June 19, 2019.

Problem Chapter 7.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y,z \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( {\frac {-b\beta \,x+2\,\arctan \left ( {{\rm e}^{\beta \,y}} \right ) a}{\beta \,b}},{\frac {-c\gamma \,x+2\,\arctan \left ( {{\rm e}^{\gamma \,z}} \right ) a}{c\gamma }} \right ) a\lambda +qx\lambda +\sinh \left ( x\lambda \right ) p \right ) }\]

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6.7.8.6 [1644] Problem 6

problem number 1644

Added June 19, 2019.

Problem Chapter 7.4.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {\cosh ^{-\text {n1}}(\text {lambda1} K[3]) \left (\text {c2} \cosh ^{\text {k2}}(\text {gamma2} K[3])+\text {b2} \cosh ^{\text {m2}}(\text {beta2} K[3])+\text {a2} \cosh ^{\text {n2}}(\text {lambda1} K[3])\right )}{\text {a1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b1} \cosh ^{\text {m1}}(\text {beta1} K[1]) \cosh ^{-\text {n1}}(\text {lambda1} K[1])}{\text {a1}}dK[1],z-\int _1^x\frac {\text {c1} \cosh ^{\text {k1}}(\text {gamma1} K[2]) \cosh ^{-\text {n1}}(\text {lambda1} K[2])}{\text {a1}}dK[2]\right )\right \}\right \}\]

Maple ✗

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

time expired

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