6.7.16 6.3
6.7.16.1 [1685] Problem 1
problem number 1685
Added June 26, 2019.
Problem Chapter 7.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \tan ^k(\lambda x)+s \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + c*D[w[x,y,z],z]== c*Tan[lambda*x]^k+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)+\frac {c \tan ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*tan(lambda*x)^k+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \tan \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]
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6.7.16.2 [1686] Problem 2
problem number 1686
Added June 26, 2019.
Problem Chapter 7.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \tan (\beta z) w_z = k \tan (\lambda x)+ s \tan (\gamma y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Tan[beta*z]*D[w[x,y,z],z]== k*Tan[lambda*x]+s*Tan[gamma*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log (\sin (\beta z))}{\beta }-\frac {c x}{a}\right )-\frac {k \log (\cos (\lambda x))}{a \lambda }-\frac {s \log (\cos (\gamma y))}{b \gamma }\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*tan(beta*z)*diff(w(x,y,z),z)= k*tan(lambda*x)+s*tan(gamma*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {2 f_{1} \left (\frac {a y -x b}{a}, \frac {-x c \beta +\ln \left (\operatorname {csgn}\left (\sec \left (\beta z \right )\right ) \sin \left (\beta z \right )\right ) a}{c \beta }\right ) a \lambda \gamma b +s \ln \left (\sec \left (\gamma y \right )^{2}\right ) a \lambda +k \ln \left (\sec \left (\lambda x \right )^{2}\right ) \gamma b}{2 a \lambda \gamma b}\]
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6.7.16.3 [1687] Problem 3
problem number 1687
Added June 26, 2019.
Problem Chapter 7.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \tan ^n(\beta x) w_y + b \tan ^k(\lambda x) w_z = c \tan ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Tan[beta*x]^n*D[w[x, y,z], y] + b*Tan[lambda*x]^k*D[w[x,y,z],z]== c*Tan[gamma*x]^m+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a \tan ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tan ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }\right )+\frac {c \tan ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*tan(beta*x)^n*diff(w(x,y,z),y)+ b*tan(lambda*x)^k*diff(w(x,y,z),z)= c*tan(gamma*x)^m+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \tan \left (\gamma x \right )^{m}d x +s x +f_{1} \left (-a \int \tan \left (\beta x \right )^{n}d x +y , -b \int \tan \left (\lambda x \right )^{k}d x +z \right )\]
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6.7.16.4 [1688] Problem 4
problem number 1688
Added June 26, 2019.
Problem Chapter 7.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \tan ^n(\lambda x) w_y + b \tan ^m(\beta y) w_z = c \tan ^k(\gamma y)+s \tan ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Tan[lambda*x]^n*D[w[x, y,z], y] + b*Tan[beta*x]^m*D[w[x,y,z],z]== c*Tan[gamma*y]^k+s*Tan[mu*z]^r;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \tan ^k\left (\frac {\gamma \left (-a \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda x)\right ) \tan ^{n+1}(\lambda x)+a \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda K[1])\right ) \tan ^{n+1}(\lambda K[1])+\lambda (n+1) y\right )}{\lambda (n+1)}\right )+s \tan ^r\left (\frac {\mu \left (-b \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\beta x)\right ) \tan ^{m+1}(\beta x)+b \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\beta K[1])\right ) \tan ^{m+1}(\beta K[1])+\beta (m+1) z\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (z-\frac {b \tan ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \tan ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*tan(lambda*x)^n*diff(w(x,y,z),y)+ b*tan(beta*x)^m*diff(w(x,y,z),z)= c*tan(gamma*y)^k+s*tan(mu*z)^r;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \tan \left (\lambda x \right )^{n}d x +y , -b \int \tan \left (\beta x \right )^{m}d x +z \right )+\int _{}^{x}\left (c {\left (\frac {-\tan \left (\gamma \left (a \int \tan \left (\lambda x \right )^{n}d x -y \right )\right )+\tan \left (\gamma a \int \tan \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} \right )}{1+\tan \left (\gamma \left (a \int \tan \left (\lambda x \right )^{n}d x -y \right )\right ) \tan \left (\gamma a \int \tan \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} \right )}\right )}^{k}+s {\left (\frac {-\tan \left (\mu \left (b \int \tan \left (\beta x \right )^{m}d x -z \right )\right )+\tan \left (\mu b \int \tan \left (\beta \textit {\_f} \right )^{m}d \textit {\_f} \right )}{1+\tan \left (\mu \left (b \int \tan \left (\beta x \right )^{m}d x -z \right )\right ) \tan \left (\mu b \int \tan \left (\beta \textit {\_f} \right )^{m}d \textit {\_f} \right )}\right )}^{r}\right )d \textit {\_f}\]
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6.7.16.5 [1689] Problem 5
problem number 1689
Added June 26, 2019.
Problem Chapter 7.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \tan ^{n_1}(\lambda _1 x) w_x + b_1 \tan ^{m_1}(\beta _1 y) w_y + c_1 \tan ^{k_1}(\gamma _1 z) w_z = a_2 \tan ^{n_2}(\lambda _2 x) + b_2 \tan ^{m_2}(\beta _2 y)+ c_2 \tan ^{k_2}(\gamma _2 z) \]
Mathematica ✗
ClearAll["Global`*"];
pde = a1*Tan[lambda1*z]^n1*D[w[x, y,z], x] + b1*Tan[beta1*y]^m1*D[w[x, y,z], y] + c1*Tan[gamma1*z]^k1*D[w[x,y,z],z]==a2*Tan[lambda2*z]^n2+ b2*Tan[beta2*y]^m2 + c2*Tan[gamma2*z]^k2;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a1*tan(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*tan(beta1*y)^m1*diff(w(x,y,z),y)+ c1*tan(gamma1*z)^k1*diff(w(x,y,z),z)= a2*tan(lambda2*x)^n2+ b2*tan(beta2*y)^m2+ c2*tan(gamma2*z)^k2;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-\int \tan \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \tan \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y}{\operatorname {b1}}, -\int \tan \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \tan \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}\right )+\frac {\int _{}^{x}\tan \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}} \left (\operatorname {a2} \tan \left (\lambda \operatorname {2} \textit {\_f} \right )^{\operatorname {n2}}+{\tan \left (\beta \operatorname {2} \operatorname {RootOf}\left (\int \tan \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {b1} -\int \tan \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {b1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\tan \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} +\operatorname {a1} \int \tan \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \right )\right )}^{\operatorname {m2}} \operatorname {b2} +{\tan \left (\gamma \operatorname {2} \operatorname {RootOf}\left (\int \tan \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {c1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\tan \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} -\int \tan \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {c1} +\operatorname {a1} \int \tan \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z \right )\right )}^{\operatorname {k2}} \operatorname {c2} \right )d \textit {\_f}}{\operatorname {a1}}\]
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