6.8.7 4.1
6.8.7.1 [1805] Problem 1
problem number 1805
Added Oct 10, 2019.
Problem Chapter 8.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \sinh ^n(\beta x) w \]
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*Sinh[beta*x]^n*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]
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*sinh(beta*x)^n*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{c \int \sinh \left (\beta x \right )^{n}d x}\]
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6.8.7.2 [1806] Problem 2
problem number 1806
Added Oct 10, 2019.
Problem Chapter 8.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \sinh (\lambda x) w_z = \left ( k \sinh (\beta x)+s \sinh (\gamma z) \right ) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Sinh[lambda*x]*D[w[x,y,z],z]== (k*Sinh[beta*x]+s*Sinh[gamma*z])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c \cosh (\lambda x)}{a \lambda }\right ) \exp \left (\int _1^x\frac {s \sinh \left (\frac {\gamma (a \lambda z-c \cosh (\lambda x)+c \cosh (\lambda K[1]))}{a \lambda }\right )+k \sinh (\beta K[1])}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x, y,z), x) + b*diff(w(x, y,z), y) + c*sinh(lambda*x)*diff(w(x,y,z),z)= (k*sinh(beta*x)+s*sinh(gamma*z))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a -b x}{a}, \frac {z a \lambda -c \cosh \left (\lambda x \right )}{a \lambda }\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (k \sinh \left (\beta \textit {\_a} \right )+s \sinh \left (\frac {\gamma \left (z a \lambda -c \cosh \left (\lambda x \right )+c \cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )\right )d \textit {\_a}}{a}}\]
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6.8.7.3 [1807] Problem 3
problem number 1807
Added Oct 10, 2019.
Problem Chapter 8.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sinh ^n(\beta x) w_y + b \sinh ^k(\lambda x) w_z = c \sinh ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Sinh[beta*x]^n*D[w[x, y,z], y] + b*Sinh[lambda*x]^k*D[w[x,y,z],z]== c*Sinh[gamma*x]^m *w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {\cosh ^2(\gamma x)} \text {sech}(\gamma x) \sinh ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-\sinh ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (y-\frac {a \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \sqrt {\cosh ^2(\lambda x)} \text {sech}(\lambda x) \sinh ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},-\sinh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \sinh \left (\beta x \right )^{n}d x +y , -b \int \sinh \left (\lambda x \right )^{k}d x +z \right ) {\mathrm e}^{c \int \sinh \left (\gamma x \right )^{m}d x}\]
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6.8.7.4 [1808] Problem 4
problem number 1808
Added Oct 10, 2019.
Problem Chapter 8.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sinh (\beta y) w_y + c \sinh (\lambda x) w_z = k \sinh (\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*Sinh[beta*y]*D[w[x, y,z], y] + c*Sinh[lambda*x]*D[w[x,y,z],z]== k*Sinh[gamma*z] *w[x,y,z];
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 (-\frac {b x}{a}-\frac {\text {arctanh}(\cosh (\beta y))}{\beta },z-\frac {c \cosh (\lambda x)}{a \lambda }\right ) \exp \left (\int _1^x\frac {k \sinh \left (\frac {\gamma (a \lambda z-c \cosh (\lambda x)+c \cosh (\lambda K[1]))}{a \lambda }\right )}{a}dK[1]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cosh (\beta y))}{\beta },z-\frac {c \cosh (\lambda x)}{a \lambda }\right ) \exp \left (\int _1^x\frac {k \sinh \left (\frac {\gamma (a \lambda z-c \cosh (\lambda x)+c \cosh (\lambda K[2]))}{a \lambda }\right )}{a}dK[2]\right )\right \}\\\end{align*}
Maple ✓
restart;
local gamma;
pde := a*diff(w(x, y,z), x) + b*sinh(beta*y)*diff(w(x, y,z), y) + c*sinh(lambda*x)*diff(w(x,y,z),z)= k*sinh(gamma*z) *w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {-x b \beta -2 \,\operatorname {arctanh}\left ({\mathrm e}^{\beta y}\right ) a}{b \beta }, \frac {z a \lambda -c \cosh \left (\lambda x \right )}{a \lambda }\right ) {\mathrm e}^{\frac {k \int _{}^{x}\sinh \left (\frac {\gamma \left (z a \lambda -c \cosh \left (\lambda x \right )+c \cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )d \textit {\_a}}{a}}\]
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6.8.7.5 [1809] Problem 5
problem number 1809
Added Oct 10, 2019.
Problem Chapter 8.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \sinh ^{n_1}(\lambda _1 x) w_x + b_1 \sinh ^{m_1}(\beta _1 y) w_y + c_1 \sinh ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \sinh ^{n_2}(\lambda _2 x) w_x + b_2 \sinh ^{m_2}(\beta _2 y) w_y + c_2 \sinh ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"];
pde = a1*Sinh[lambda1*x]^n1*D[w[x, y,z], x] + b1*Sinh[beta1*y]^m1*D[w[x, y,z], y] + c1*Sinh[gamma1*x]^k1*D[w[x, y,z], z]== (a2*Sinh[lambda2*x]^n2+b2*Sinh[beta2*y]^m2+c2*Sinh[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*sinh(lambda1*x)^n1*diff(w(x, y,z), x) + b1*sinh(beta1*y)^m1*diff(w(x, y,z), y) + c1*sinh(gamma1*x)^k1*diff(w(x,y,z),z)= ( a2*sinh(lambda2*x)^n2+b2*sinh(beta2*y)^m2+c2*sinh(gamma2*x)^k2) *w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-\int \sinh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \sinh \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y}{\operatorname {b1}}, -\frac {\operatorname {c1} \int \sinh \left (\gamma \operatorname {1} x \right )^{\operatorname {k1}} \sinh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x}{\operatorname {a1}}+z \right ) {\mathrm e}^{\frac {\int _{}^{x}\left (\operatorname {a2} \sinh \left (\lambda \operatorname {2} \textit {\_f} \right )^{\operatorname {n2}}+\operatorname {b2} {\sinh \left (\beta \operatorname {2} \operatorname {RootOf}\left (\int \sinh \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {b1} +\operatorname {a1} \int \sinh \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y -\int \sinh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {b1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\sinh \left (\beta \operatorname {1} \textit {\_a} \right )^{-\operatorname {m1}}d \textit {\_a} \right )\right )}^{\operatorname {m2}}+\operatorname {c2} \sinh \left (\gamma \operatorname {2} \textit {\_f} \right )^{\operatorname {k2}}\right ) \sinh \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f}}{\operatorname {a1}}}\]
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