6.7.14 6.1
6.7.14.1 [1673] Problem 1
problem number 1673
Added June 26, 2019.
Problem Chapter 7.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \sin ^k(\lambda x)+s \]
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*Sin[lambda*x]^k+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)+\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*sin(lambda*x)^k+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \sin \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]
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6.7.14.2 [1674] Problem 2
problem number 1674
Added June 26, 2019.
Problem Chapter 7.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \sin (\gamma z) w_z = k \sin (\alpha x)+ s \sin (\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Sin[gamma*z]*D[w[x,y,z],z]== k*Sin[alpha*x]+s*Sin[beta*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\frac {c x}{a}-\frac {\text {arctanh}(\cos (\gamma z))}{\gamma }\right )-\frac {k \cos (\alpha x)}{a \alpha }-\frac {s \cos (\beta y)}{b \beta }\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*sin(gamma*z)*diff(w(x,y,z),z)= k*sin(alpha*x)+s*sin(beta*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {f_{1} \left (\frac {x c \gamma +a \ln \left (\csc \left (\gamma z \right )+\cot \left (\gamma z \right )\right )}{c \gamma }, \frac {y c \gamma +b \ln \left (\csc \left (\gamma z \right )+\cot \left (\gamma z \right )\right )}{c \gamma }\right ) \alpha a \beta b -k \cos \left (\alpha x \right ) \beta b -s \cos \left (\beta y \right ) \alpha a}{\alpha a \beta b}\]
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6.7.14.3 [1675] Problem 3
problem number 1675
Added June 26, 2019.
Problem Chapter 7.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta x) w_z = c \sin ^k(\gamma x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*x]^m*D[w[x,y,z],z]== c*Sin[gamma*x]^k;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\gamma x)\right )}{\gamma k+\gamma }+c_1\left (z-\frac {b \sqrt {\cos ^2(\beta x)} \sec (\beta x) \sin ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*x)^m*diff(w(x,y,z),z)= c*sin(gamma*x)^k;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \sin \left (\gamma x \right )^{k}d x +f_{1} \left (-a \int \sin \left (\lambda x \right )^{n}d x +y , -b \int \sin \left (\beta x \right )^{m}d x +z \right )\]
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6.7.14.4 [1676] Problem 4
problem number 1676
Added June 26, 2019.
Problem Chapter 7.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta y) w_z = c \sin ^k(\gamma y)+s \sin ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*x]^m*D[w[x,y,z],z]== c*Sin[gamma*y]^k+s*Sin[mu*z]^r;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \sin ^k\left (\frac {\gamma \left (-a \sqrt {\cos ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right ) \sec (\lambda x) \sin ^{n+1}(\lambda x)+a \sqrt {\cos ^2(\lambda K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda K[1])\right ) \sec (\lambda K[1]) \sin ^{n+1}(\lambda K[1])+\lambda (n+1) y\right )}{\lambda (n+1)}\right )+s \sin ^r\left (\frac {\mu \left (-b \sqrt {\cos ^2(\beta x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\beta x)\right ) \sec (\beta x) \sin ^{m+1}(\beta x)+b \sqrt {\cos ^2(\beta K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\beta K[1])\right ) \sec (\beta K[1]) \sin ^{m+1}(\beta K[1])+\beta (m+1) z\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (z-\frac {b \sqrt {\cos ^2(\beta x)} \sec (\beta x) \sin ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*x)^m*diff(w(x,y,z),z)= c*sin(gamma*y)^k+s*sin(mu*z)^r;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \sin \left (\lambda x \right )^{n}d x +y , -b \int \sin \left (\beta x \right )^{m}d x +z \right )+\int _{}^{x}\left (c {\left (-\sin \left (\gamma \left (a \int \sin \left (\lambda x \right )^{n}d x -a \int \sin \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -y \right )\right )\right )}^{k}+s {\left (-\sin \left (\mu \left (b \int \sin \left (\beta x \right )^{m}d x -b \int \sin \left (\beta \textit {\_f} \right )^{m}d \textit {\_f} -z \right )\right )\right )}^{r}\right )d \textit {\_f}\]
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6.7.14.5 [1677] Problem 5
problem number 1677
Added June 26, 2019.
Problem Chapter 7.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin (\beta x) w_y + c \sin (\lambda x) w_z = k \sin (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*Sin[beta*x]*D[w[x, y,z], y] + c*Sin[lambda*x]*D[w[x,y,z],z]== k*Sin[gamma*z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \sin \left (\gamma \left (z-\int _1^x\frac {c \sin (\lambda K[2])}{a}dK[2]+\int _1^{K[3]}\frac {c \sin (\lambda K[2])}{a}dK[2]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \sin (\beta K[1])}{a}dK[1],z-\int _1^x\frac {c \sin (\lambda K[2])}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*sin(beta*x)*diff(w(x,y,z),y)+ c*sin(lambda*x)*diff(w(x,y,z),z)= k*sin(gamma*z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a \beta +b \cos \left (\beta x \right )}{a \beta }, \frac {z a \lambda +c \cos \left (\lambda x \right )}{a \lambda }\right )-\frac {k \int _{}^{x}-\sin \left (\frac {\gamma \left (z a \lambda +c \cos \left (\lambda x \right )-c \cos \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )d \textit {\_a}}{a}\]
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6.7.14.6 [1678] Problem 6
problem number 1678
Added June 26, 2019.
Problem Chapter 7.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \sin ^{n_1}(\lambda _1 x) w_x + b_1 \sin ^{m_1}(\beta _1 y) w_y + c_1 \sin ^{k_1}(\gamma _1 z) w_z = a_2 \sin ^{n_2}(\lambda _2 x) + b_2 \sin ^{m_2}(\beta _2 y)+ c_2 \sin ^{k_2}(\gamma _2 z) \]
Mathematica ✗
ClearAll["Global`*"];
pde = a1*Sin[lambda1*z]^n1*D[w[x, y,z], x] + b1*Sin[beta1*y]^m1*D[w[x, y,z], y] + c1*Sin[gamma1*z]^k1*D[w[x,y,z],z]==a2*Sin[lambda2*z]^n2+ b2*Sin[beta2*y]^m2 + c2*Sin[gamma2*z]^k2;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a1*sin(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*sin(beta1*y)^m1*diff(w(x,y,z),y)+ c1*sin(gamma1*z)^k1*diff(w(x,y,z),z)= a2*sin(lambda2*x)^n2+ b2*sin(beta2*y)^m2+ c2*sin(gamma2*z)^k2;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-\int \sin \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \sin \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y}{\operatorname {b1}}, -\int \sin \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \sin \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}\right )+\frac {\int _{}^{x}\sin \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}} \left (\operatorname {a2} \sin \left (\lambda \operatorname {2} \textit {\_f} \right )^{\operatorname {n2}}+{\sin \left (\beta \operatorname {2} \operatorname {RootOf}\left (\operatorname {a1} \int \sin \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y -\operatorname {a1} \int _{}^{\textit {\_Z}}\sin \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} +\int \sin \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {b1} -\int \sin \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {b1} \right )\right )}^{\operatorname {m2}} \operatorname {b2} +\operatorname {c2} {\sin \left (\gamma \operatorname {2} \operatorname {RootOf}\left (\int \sin \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {c1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\sin \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} -\int \sin \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {c1} +\operatorname {a1} \int \sin \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z \right )\right )}^{\operatorname {k2}}\right )d \textit {\_f}}{\operatorname {a1}}\]
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