6.6.8 4.2

6.6.8.1 [1470] Problem 1

problem number 1470

Added May 19, 2019.

Problem Chapter 6.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \cosh (\beta x) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*Cosh[beta*x]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*cosh(beta*x)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a -b x}{a}, \frac {z a \beta -c \sinh \left (\beta x \right )}{a \beta }\right )\]

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6.6.8.2 [1471] Problem 2

problem number 1471

Added May 19, 2019.

Problem Chapter 6.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cosh (\beta x) w_y + c \cosh (\lambda x) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Cosh[beta*x]*D[w[x, y,z], y] +c*Cosh[lambda*x]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*cosh(beta*x)*diff(w(x,y,z),y)+c*cosh(lambda*x)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a \beta -b \sinh \left (\beta x \right )}{a \beta }, \frac {z a \lambda -c \sinh \left (\lambda x \right )}{a \lambda }\right )\]

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6.6.8.3 [1472] Problem 3

problem number 1472

Added May 19, 2019.

Problem Chapter 6.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cosh (\beta y) w_y + c \cosh (\lambda x) w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.8.4 [1473] Problem 4

problem number 1473

Added May 19, 2019.

Problem Chapter 6.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cosh (\beta y) w_y + c \cosh (\gamma z) w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {\cot ^{-1}(\sinh (\beta y))}{\beta },-\frac {c x}{a}-\frac {\cot ^{-1}(\sinh (\gamma z))}{\gamma }\right )\right \}\right \}\]

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.8.5 [1474] Problem 5

problem number 1474

Added May 19, 2019.

Problem Chapter 6.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \cosh (\lambda x) w_x + b \cosh (\beta y) w_y + c \cosh (\gamma z) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Cosh[lambda*x]*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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \cot ^{-1}(\sinh (\lambda x))}{a \lambda }-\frac {\cot ^{-1}(\sinh (\beta y))}{\beta },\frac {c \cot ^{-1}(\sinh (\lambda x))}{a \lambda }-\frac {\cot ^{-1}(\sinh (\gamma z))}{\gamma }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*cosh(lambda*x)*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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.8.6 [1475] Problem 6

problem number 1475

Added May 19, 2019.

Problem Chapter 6.4.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \cosh (\beta y) w_x + b \cosh (\lambda x) w_y + c \cosh (\gamma z) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Cosh[beta*y]*D[w[x, y,z], x] + b*Cosh[lambda*x]*D[w[x, y,z], y] +c*Cosh[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\begin{align*}& \left \{w(x,y,z)\to c_1\left (\int _1^y\cosh (\beta K[1])dK[1]-\int _1^x\frac {b \cosh (\lambda K[2])}{a}dK[2],-\frac {\gamma \int _1^x\frac {c \text {sech}\left (\beta \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\cosh (\beta K[1])dK[1]\&\right ]\left [\int _1^y\cosh (\beta K[1])dK[1]-\int _1^x\frac {b \cosh (\lambda K[2])}{a}dK[2]+\int _1^{K[3]}\frac {b \cosh (\lambda K[2])}{a}dK[2]\right ]\right )}{a}dK[3]+\cot ^{-1}(\sinh (\gamma z))}{\gamma }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\int _1^y\cosh (\beta K[1])dK[1]-\int _1^x\frac {b \cosh (\lambda K[2])}{a}dK[2],\frac {\cot ^{-1}(\sinh (\gamma z))}{\gamma }-\int _1^x\frac {c \text {sech}\left (\beta \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\cosh (\beta K[1])dK[1]\&\right ]\left [\int _1^y\cosh (\beta K[1])dK[1]-\int _1^x\frac {b \cosh (\lambda K[2])}{a}dK[2]+\int _1^{K[3]}\frac {b \cosh (\lambda K[2])}{a}dK[2]\right ]\right )}{a}dK[3]\right )\right \}\\\end{align*}

Maple ✓

restart; 
pde :=  a*cosh(beta*y)*diff(w(x,y,z),x)+ b*cosh(lambda*x)*diff(w(x,y,z),y)+c*cosh(gamma*z)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[\text {Expression too large to display}\]

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