6.7.23 7.4
6.7.23.1 [1717] Problem 1
problem number 1717
Added June 26, 2019.
Problem Chapter 7.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \arccot ^k(\lambda x)+s \]
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*ArcCot[lambda*x]^k+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \cot ^{-1}(\lambda K[1])^k+s\right )dK[1]+c_1(y-a x,z-b x)\right \}\right \}\]
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*arccot(lambda*x)^k+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \operatorname {arccot}\left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]
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6.7.23.2 [1718] Problem 2
problem number 1718
Added June 26, 2019.
Problem Chapter 7.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 w_x + a_2 w_y + a_3 w_z = b_1 \arccot (\lambda _1 x)+b_2 \arccot (\lambda _2 y)+b_3 \arccot (\lambda _3 z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a1*D[w[x, y,z], x] + a2*D[w[x, y,z], y] + a3*D[w[x,y,z],z]== b1*ArcCot[lambda1*x]+b2*ArcCot[lambda2*y]+b3*ArcCot[lambda3*z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right )+\frac {1}{2} \left (\frac {\text {b2} \log \left (\text {a1}^2 \left (\text {lambda2}^2 y^2+1\right )\right )}{\text {a2} \text {lambda2}}+\frac {\text {b3} \log \left (\text {a1}^2 \left (\text {lambda3}^2 z^2+1\right )\right )}{\text {a3} \text {lambda3}}+\frac {2 \text {b2} x \arctan (\text {lambda2} y)}{\text {a1}}+\frac {2 \text {b3} x \arctan (\text {lambda3} z)}{\text {a1}}+\frac {\text {b1} \log \left (\text {lambda1}^2 x^2+1\right )}{\text {a1} \text {lambda1}}+\frac {2 \text {b2} x \cot ^{-1}(\text {lambda2} y)}{\text {a1}}+\frac {2 \text {b3} x \cot ^{-1}(\text {lambda3} z)}{\text {a1}}-\frac {2 \text {b2} y \arctan (\text {lambda2} y)}{\text {a2}}-\frac {2 \text {b3} z \arctan (\text {lambda3} z)}{\text {a3}}\right )+\frac {\text {b1} x \cot ^{-1}(\text {lambda1} x)}{\text {a1}}\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a1*diff(w(x,y,z),x)+ a2*diff(w(x,y,z),y)+ a3*diff(w(x,y,z),z)= b1*arccot(lambda1*x)+b2*arccot(lambda2*y)+b3*arccot(lambda3*z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {2 f_{1} \left (\frac {\operatorname {a1} y -\operatorname {a2} x}{\operatorname {a1}}, \frac {\operatorname {a1} z -\operatorname {a3} x}{\operatorname {a1}}\right ) \operatorname {a1} \lambda \operatorname {1} \lambda \operatorname {2} \operatorname {a2} \lambda \operatorname {3} \operatorname {a3} +\operatorname {b1} \ln \left (\lambda \operatorname {1}^{2} x^{2}+1\right ) \lambda \operatorname {2} \operatorname {a2} \lambda \operatorname {3} \operatorname {a3} +2 \lambda \operatorname {1} \left (\frac {\lambda \operatorname {3} \ln \left (\lambda \operatorname {2}^{2} y^{2}+1\right ) \operatorname {a1} \operatorname {a3} \operatorname {b2}}{2}+\left (\frac {\ln \left (\lambda \operatorname {3}^{2} z^{2}+1\right ) \operatorname {a1} \operatorname {a2} \operatorname {b3}}{2}+\lambda \operatorname {3} \left (x \,\operatorname {arccot}\left (\lambda \operatorname {1} x \right ) \operatorname {a2} \operatorname {a3} \operatorname {b1} +\operatorname {a1} \left (\operatorname {arccot}\left (\lambda \operatorname {2} y \right ) y \operatorname {a3} \operatorname {b2} +\operatorname {arccot}\left (\lambda \operatorname {3} z \right ) z \operatorname {a2} \operatorname {b3} \right )\right )\right ) \lambda \operatorname {2} \right )}{2 \operatorname {a1} \lambda \operatorname {1} \lambda \operatorname {2} \operatorname {a2} \lambda \operatorname {3} \operatorname {a3}}\]
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6.7.23.3 [1719] Problem 3
problem number 1719
Added June 26, 2019.
Problem Chapter 7.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arccot ^n(\lambda x) \arccot ^k(\beta z) w_z = s \arccot ^m(\gamma x) \]
Mathematica ✗
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*ArcCot[lambda*x]^n*ArcCot[beta*z]^k*D[w[x,y,z],z]== s*ArcCot[gamma*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*arccot(lambda*x)^n*arccot(beta*z)^k*diff(w(x,y,z),z)= s*arccot(gamma*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {s \int \operatorname {arccot}\left (\gamma x \right )^{m}d x}{a}+f_{1} \left (\frac {a y -b x}{a}, -\int \operatorname {arccot}\left (\lambda x \right )^{n}d x +\frac {a \int \operatorname {arccot}\left (\beta z \right )^{-k}d z}{c}\right )\]
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6.7.23.4 [1720] Problem 4
problem number 1720
Added June 26, 2019.
Problem Chapter 7.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arccot ^n(\lambda x) \arccot ^m(\beta y) \arccot ^k(\gamma z) w_z = s \]
Mathematica ✗
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*ArcCot[lambda*x]^n*ArcCot[beta*y]^m*ArcCot[gamma*z]^k*D[w[x,y,z],z]== s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*arccot(lambda*x)^n*arccot(beta*y)^m*arccot(gamma*z)^k*diff(w(x,y,z),z)= s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {s x}{a}+f_{1} \left (\frac {a y -b x}{a}, -\int _{}^{x}\operatorname {arccot}\left (\lambda \textit {\_a} \right )^{n} \operatorname {arccot}\left (\frac {\left (a y -b \left (-\textit {\_a} +x \right )\right ) \beta }{a}\right )^{m}d \textit {\_a} +\frac {a \int \operatorname {arccot}\left (\gamma z \right )^{-k}d z}{c}\right )\]
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6.7.23.5 [1721] Problem 5
problem number 1721
Added June 26, 2019.
Problem Chapter 7.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccot ^n(\lambda x) w_y + c \arccot ^k(\beta z) w_z = s \arccot ^m(\gamma x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*ArcTan[lambda*x]^n*D[w[x, y,z], y] + c*ArcTan[beta*z]^k*D[w[x,y,z],z]== s*ArcCot[gamma*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^z\frac {s \cot ^{-1}\left (\frac {\gamma \left (c x-a \int _1^z\arctan (\beta K[2])^{-k}dK[2]+a \int _1^{K[3]}\arctan (\beta K[2])^{-k}dK[2]\right )}{c}\right ){}^m \arctan (\beta K[3])^{-k}}{c}dK[3]+c_1\left (y-\int _1^x\frac {b \arctan (\lambda K[1])^n}{a}dK[1],\int _1^z\arctan (\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*arctan(lambda*x)^n*diff(w(x,y,z),y)+ c*arctan(beta*z)^k*diff(w(x,y,z),z)= s*arccot(gamma*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {s \int _{}^{y}{\operatorname {arccot}\left (\gamma \operatorname {RootOf}\left (b \int \arctan \left (\lambda x \right )^{n}d x -b \int _{}^{\textit {\_Z}}\arctan \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} +\textit {\_b} a -y a \right )\right )}^{m} {\arctan \left (\lambda \operatorname {RootOf}\left (b \int \arctan \left (\lambda x \right )^{n}d x -b \int _{}^{\textit {\_Z}}\arctan \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} +\textit {\_b} a -y a \right )\right )}^{-n}d \textit {\_b}}{b}+f_{1} \left (-y +\frac {b \int \arctan \left (\lambda x \right )^{n}d x}{a}, -\int _{}^{y}{\arctan \left (\lambda \operatorname {RootOf}\left (b \int \arctan \left (\lambda x \right )^{n}d x -b \int _{}^{\textit {\_Z}}\arctan \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} +\textit {\_b} a -y a \right )\right )}^{-n}d \textit {\_b} +\frac {b \int \arctan \left (\beta z \right )^{-k}d z}{c}\right )\]
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