6.9.7 4.1
6.9.7.1 [1971] Problem 1
problem number 1971
Added Jan 19, 2020.
Problem Chapter 9.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \sinh ^n(\beta x) w + k \sinh ^m(\lambda x) \]
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]+ k*Sinh[lambda*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \sqrt {\cosh ^2(\beta K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta K[1])\right ) \text {sech}(\beta K[1]) \sinh ^{n+1}(\beta K[1])}{n \beta +\beta }\right ) k \sinh ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
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)+ k*sinh(lambda*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (k \int \sinh \left (\lambda x \right )^{m} {\mathrm e}^{-c \int \sinh \left (\beta x \right )^{n}d x}d x +f_{1} \left (-a x +y , -b x +z \right )\right ) {\mathrm e}^{c \int \sinh \left (\beta x \right )^{n}d x}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.9.7.2 [1972] Problem 2
problem number 1972
Added Jan 19, 2020.
Problem Chapter 9.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \sinh (\beta z) w_z = \left (p \sinh (\lambda x) + q \right ) w + k \sinh (\gamma x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x,y,z],x]+ b*D[w[x,y,z],y]+c*Sinh[beta*z]*D[w[x,y,z],z]==(p*Sinh[lambda*x]+q)*w[x,y,z]+ k*Sinh[gamma*x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\begin{align*}& \left \{w(x,y,z)\to e^{\frac {p \cosh (\lambda x)+\lambda q x}{a \lambda }} \left (\int _1^x\frac {e^{-\frac {p \cosh (\lambda K[1])+\lambda q K[1]}{a \lambda }} k \sinh (\gamma K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a},-\frac {c x}{a}-\frac {\text {arctanh}(\cosh (\beta z))}{\beta }\right )\right )\right \}\\& \left \{w(x,y,z)\to e^{\frac {p \cosh (\lambda x)+\lambda q x}{a \lambda }} \left (\int _1^x\frac {e^{-\frac {p \cosh (\lambda K[2])+\lambda q K[2]}{a \lambda }} k \sinh (\gamma K[2])}{a}dK[2]+c_1\left (y-\frac {b x}{a},-\frac {c x}{a}-\frac {\text {arctanh}(\cosh (\beta z))}{\beta }\right )\right )\right \}\\\end{align*}
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*sinh(beta*z)*diff(w(x,y,z),z)=(p*sinh(lambda*x)+q)*w(x,y,z)+ k*sinh(gamma*x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\frac {k \int \sinh \left (\gamma x \right ) {\mathrm e}^{\frac {-q x \lambda -p \cosh \left (\lambda x \right )}{a \lambda }}d x}{a}+f_{1} \left (\frac {a y -b x}{a}, \frac {-x c \beta -2 a \,\operatorname {arctanh}\left ({\mathrm e}^{\beta z}\right )}{c \beta }\right )\right ) {\mathrm e}^{\frac {q x \lambda +p \cosh \left (\lambda x \right )}{a \lambda }}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.9.7.3 [1973] Problem 3
problem number 1973
Added Jan 19, 2020.
Problem Chapter 9.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sinh ^n(\beta x) w_y + b \sinh ^k(\lambda x) w_z = c w + s \sinh ^m(\mu x) \]
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*w[x,y,z]+ k*Sinh[mu*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {k \left (e^{\mu x}-e^{-\mu x}\right )^m \left (2-2 e^{2 \mu x}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-\frac {c+m \mu }{2 \mu },-\frac {c+(m-2) \mu }{2 \mu },e^{2 \mu x}\right )}{c+m \mu }+e^{c x} c_1\left (y-\frac {a \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \sqrt {\cosh ^2(\lambda x)} \text {sech}(\lambda x) \sinh ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},-\sinh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
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*w(x,y,z)+ k*sinh(mu*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (k \int \sinh \left (\mu x \right )^{m} {\mathrm e}^{-c x}d x +f_{1} \left (-a \int \sinh \left (\beta x \right )^{n}d x +y , -b \int \sinh \left (\lambda x \right )^{k}d x +z \right )\right ) {\mathrm e}^{c x}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.9.7.4 [1974] Problem 4
problem number 1974
Added Jan 19, 2020.
Problem Chapter 9.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + b \sinh ^n(\beta x) w_y + c \sinh ^k(\lambda y) w_z = a w + s \sinh ^m(\mu x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+ b*Sinh[beta*x]^n*D[w[x,y,z],y]+c*Sinh[lambda*y]^k*D[w[x,y,z],z]==a*w[x,y,z]+ s*Sinh[mu*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {s \left (e^{\mu x}-e^{-\mu x}\right )^m \left (2-2 e^{2 \mu x}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-\frac {a+m \mu }{2 \mu },-\frac {a+(m-2) \mu }{2 \mu },e^{2 \mu x}\right )}{a+m \mu }+e^{a x} c_1\left (y-\frac {b \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta x)\right )}{\beta n+\beta },z-\int _1^xc \sinh ^k\left (\frac {\lambda \left (-b \sqrt {\cosh ^2(\beta x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta x)\right ) \text {sech}(\beta x) \sinh ^{n+1}(\beta x)+b \sqrt {\cosh ^2(\beta K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\beta K[1])\right ) \text {sech}(\beta K[1]) \sinh ^{n+1}(\beta K[1])+\beta (n+1) y\right )}{\beta (n+1)}\right )dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ b*sinh(beta*x)^n*diff(w(x,y,z),y)+ c*sinh(lambda*y)^k*diff(w(x,y,z),z)=a*w(x,y,z)+ s*sinh(mu*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (s \int \sinh \left (\mu x \right )^{m} {\mathrm e}^{-a x}d x +f_{1} \left (-b \int \sinh \left (\beta x \right )^{n}d x +y , -c \int _{}^{x}{\sinh \left (\lambda \left (b \int \sinh \left (\beta \textit {\_b} \right )^{n}d \textit {\_b} -b \int \sinh \left (\beta x \right )^{n}d x +y \right )\right )}^{k}d \textit {\_b} +z \right )\right ) {\mathrm e}^{a x}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.9.7.5 [1975] Problem 5
problem number 1975
Added Jan 19, 2020.
Problem Chapter 9.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b_1 \sinh ^{n_1}(\lambda _1 x) w_x + b_2 \sinh ^{n_2}(\lambda _2 y) w_y + b_3 \sinh ^{n_3}(\lambda _3 z) w_z = a w + c_1 \sinh ^{k_1}(\beta _1 x)+ c_2 \sinh ^{k_2}(\beta _2 y)+ c_3 \sinh ^{k_3}(\beta _3 z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = b1*Sinh[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Sinh[lambda2*x]^n2*D[w[x,y,z],y]+b3*Sinh[lambda3*x]^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Sinh[beta1*x]^k1+ c2*Sinh[beta2*x]^k2+ c3*Sinh[beta3*x]^k3;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} x)} \text {sech}(\text {lambda1} x) \sinh ^{1-\text {n1}}(\text {lambda1} x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},-\sinh ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (\int _1^x\frac {\exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} K[3])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},-\sinh ^2(\text {lambda1} K[3])\right ) \text {sech}(\text {lambda1} K[3]) \sinh ^{1-\text {n1}}(\text {lambda1} K[3])}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \sinh ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \sinh ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \sinh ^{\text {k3}}(\text {beta3} K[3])\right ) \sinh ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b2} \sinh ^{-\text {n1}}(\text {lambda1} K[1]) \sinh ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \sinh ^{-\text {n1}}(\text {lambda1} K[2]) \sinh ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := b__1*sinh(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*sinh(lambda__2*x)^(n__2)*diff(w(x,y,z),y)+ b__3*sinh(lambda__3*x)^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*sinh(beta__1*x)^(k__1)+ c__2*sinh(beta__2*x)^(k__2)+ c__3*sinh(beta__3*x)^(k__3);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\frac {\int \left (c_{1} \sinh \left (\beta _{1} x \right )^{k_{1}}+c_{2} \sinh \left (\beta _{2} x \right )^{k_{2}}+c_{3} \sinh \left (\beta _{3} x \right )^{k_{3}}\right ) \sinh \left (\lambda _{1} x \right )^{-n_{1}} {\mathrm e}^{-\frac {a \int \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}}d x}{b_{1}}+f_{1} \left (-\frac {b_{2} \int \sinh \left (\lambda _{2} x \right )^{n_{2}} \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}+y , -\frac {b_{3} \int \sinh \left (\lambda _{3} x \right )^{n_{3}} \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}+z \right )\right ) {\mathrm e}^{\frac {a \int \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________