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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = c \int \cosh \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]

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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]];
 

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

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

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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

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

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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]];
 

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

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'));
 

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

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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'));
 

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

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