6.7.15 6.2
6.7.15.1 [1679] Problem 1
problem number 1679
Added June 26, 2019.
Problem Chapter 7.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \cos ^k(\lambda x)+s \]
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*Cos[lambda*x]^k+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)-\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cos ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*cos(lambda*x)^k+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \cos \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]
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6.7.15.2 [1680] Problem 2
problem number 1680
Added June 26, 2019.
Problem Chapter 7.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \cos (\beta z) w_z = k \cos (\lambda x)+ s \cos (\gamma y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Cos[beta*z]*D[w[x,y,z],z]== k*Cos[lambda*x]+s*Cos[gamma*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\coth ^{-1}(\sin (\beta z))}{\beta }-\frac {c x}{a}\right )+\frac {k \sin (\lambda x)}{a \lambda }+\frac {s \sin (\gamma y)}{b \gamma }\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*cos(beta*z)*diff(w(x,y,z),z)= k*cos(lambda*x)+s*cos(gamma*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {f_{1} \left (\frac {x c \beta -a \ln \left (\sec \left (\beta z \right )+\tan \left (\beta z \right )\right )}{c \beta }, \frac {y c \beta -b \ln \left (\sec \left (\beta z \right )+\tan \left (\beta z \right )\right )}{c \beta }\right ) \lambda a \gamma b +k \sin \left (\lambda x \right ) \gamma b +s \sin \left (\gamma y \right ) \lambda a}{\lambda a \gamma b}\]
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6.7.15.3 [1681] Problem 3
problem number 1681
Added June 26, 2019.
Problem Chapter 7.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cos ^n(\beta x) w_y + b \cos ^k(\lambda x) w_z = c \cos ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Cos[beta*x]^n*D[w[x, y,z], y] + b*Cos[lambda*x]^k*D[w[x,y,z],z]== c*Cos[gamma*x]^m+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\beta x)\right )}{\beta n+\beta }+y,\frac {b \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cos ^2(\lambda x)\right )}{k \lambda +\lambda }+z\right )-\frac {c \sqrt {\sin ^2(\gamma x)} \csc (\gamma x) \cos ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*cos(beta*x)^n*diff(w(x,y,z),y)+ b*cos(lambda*x)^k*diff(w(x,y,z),z)= c*cos(gamma*x)^m+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \cos \left (\gamma x \right )^{m}d x +s x +f_{1} \left (-a \int \cos \left (\beta x \right )^{n}d x +y , -b \int \cos \left (\lambda x \right )^{k}d x +z \right )\]
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6.7.15.4 [1682] Problem 4
problem number 1682
Added June 26, 2019.
Problem Chapter 7.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cos ^n(\lambda x) w_y + b \cos ^m(\beta y) w_z = c \cos ^k(\gamma y)+s \cos ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Cos[lambda*x]^n*D[w[x, y,z], y] + b*Cos[beta*x]^m*D[w[x,y,z],z]== c*Cos[gamma*y]^k+s*Cos[mu*z]^r;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \cos ^k\left (\frac {\gamma \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )+s \cos ^r\left (\frac {\mu \left (b \csc (\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\beta x)\right ) \sqrt {\sin ^2(\beta x)} \cos ^{m+1}(\beta x)+\beta (m+1) z-b \cos ^{m+1}(\beta K[1]) \csc (\beta K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\beta K[1])\right ) \sqrt {\sin ^2(\beta K[1])}\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\beta x)\right )}{\beta m+\beta }+z,\frac {a \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right )}{\lambda n+\lambda }+y\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*cos(lambda*x)^n*diff(w(x,y,z),y)+ b*cos(beta*x)^m*diff(w(x,y,z),z)= c*cos(gamma*y)^k+s*cos(mu*z)^r;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \cos \left (\lambda x \right )^{n}d x +y , -b \int \cos \left (\beta x \right )^{m}d x +z \right )+\int _{}^{x}\left (c {\cos \left (\gamma \left (a \int \cos \left (\lambda x \right )^{n}d x -a \int \cos \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -y \right )\right )}^{k}+s {\cos \left (\mu \left (b \int \cos \left (\beta \textit {\_f} \right )^{m}d \textit {\_f} -b \int \cos \left (\beta x \right )^{m}d x +z \right )\right )}^{r}\right )d \textit {\_f}\]
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6.7.15.5 [1683] Problem 5
problem number 1683
Added June 26, 2019.
Problem Chapter 7.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cos (\beta x) w_y + c \cos (\lambda x) w_z = k \cos (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*Cos[beta*x]*D[w[x, y,z], y] + c*Cos[lambda*x]*D[w[x,y,z],z]== k*Cos[gamma*z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \cos \left (\frac {\gamma (a \lambda z-c \sin (\lambda x)+c \sin (\lambda K[1]))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b \sin (\beta x)}{a \beta },z-\frac {c \sin (\lambda x)}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*cos(beta*x)*diff(w(x,y,z),y)+ c*cos(lambda*x)*diff(w(x,y,z),z)= k*cos(gamma*z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a \beta -b \sin \left (\beta x \right )}{a \beta }, \frac {z a \lambda -c \sin \left (\lambda x \right )}{a \lambda }\right )+\frac {k \int _{}^{x}\cos \left (\frac {\gamma \left (z a \lambda -c \sin \left (\lambda x \right )+c \sin \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )d \textit {\_a}}{a}\]
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6.7.15.6 [1684] Problem 6
problem number 1684
Added June 26, 2019.
Problem Chapter 7.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \cos ^{n_1}(\lambda _1 x) w_x + b_1 \cos ^{m_1}(\beta _1 y) w_y + c_1 \cos ^{k_1}(\gamma _1 z) w_z = a_2 \cos ^{n_2}(\lambda _2 x) + b_2 \cos ^{m_2}(\beta _2 y)+ c_2 \cos ^{k_2}(\gamma _2 z) \]
Mathematica ✗
ClearAll["Global`*"];
pde = a1*Cos[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cos[beta1*y]^m1*D[w[x, y,z], y] + c1*Cos[gamma1*z]^k1*D[w[x,y,z],z]==a2*Cos[lambda2*z]^n2+ b2*Cos[beta2*y]^m2 + c2*Cos[gamma2*z]^k2;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a1*cos(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*cos(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cos(gamma1*z)^k1*diff(w(x,y,z),z)= a2*cos(lambda2*x)^n2+ b2*cos(beta2*y)^m2+ c2*cos(gamma2*z)^k2;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-\int \cos \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \cos \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y}{\operatorname {b1}}, -\int \cos \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \cos \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}\right )+\frac {\int _{}^{x}\cos \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}} \left (\operatorname {a2} \cos \left (\lambda \operatorname {2} \textit {\_f} \right )^{\operatorname {n2}}+{\cos \left (\beta \operatorname {2} \operatorname {RootOf}\left (\operatorname {a1} \int \cos \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y -\operatorname {a1} \int _{}^{\textit {\_Z}}\cos \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} +\int \cos \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {b1} -\int \cos \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {b1} \right )\right )}^{\operatorname {m2}} \operatorname {b2} +{\cos \left (\gamma \operatorname {2} \operatorname {RootOf}\left (\int \cos \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {c1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\cos \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} -\int \cos \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {c1} +\operatorname {a1} \int \cos \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z \right )\right )}^{\operatorname {k2}} \operatorname {c2} \right )d \textit {\_f}}{\operatorname {a1}}\]
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