6.8.14 6.1
6.8.14.1 [1842] Problem 1
problem number 1842
Added Oct 18, 2019.
Problem Chapter 8.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \sin ^n(\lambda x) w \]
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*Sin[lambda*x]^n*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)= c*sin(lambda*x)^n*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{c \int \sin \left (\lambda x \right )^{n}d x}\]
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6.8.14.2 [1843] Problem 2
problem number 1843
Added Oct 18, 2019.
Problem Chapter 8.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \sin (\lambda z) w_z = \left ( k \sin (\gamma x)+s \sin (\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Sin[lambda*z]*D[w[x,y,z],z]== (k*Sin[gamma*x]+s*Sin[beta*y])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{-\frac {k \cos (\gamma x)}{a \gamma }-\frac {s \cos (\beta y)}{b \beta }} c_1\left (y-\frac {b x}{a},-\frac {c x}{a}-\frac {\text {arctanh}(\cos (\lambda z))}{\lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+ c*sin(lambda*z)*diff(w(x,y,z),z)= (k*sin(gamma*x)+s*sin(beta*y))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {x c \lambda +\ln \left (\csc \left (\lambda z \right )+\cot \left (\lambda z \right )\right ) a}{c \lambda }, \frac {y c \lambda +b \ln \left (\csc \left (\lambda z \right )+\cot \left (\lambda z \right )\right )}{c \lambda }\right ) {\mathrm e}^{\frac {-k \cos \left (\gamma x \right ) b \beta -s \cos \left (\beta y \right ) \gamma a}{\gamma a b \beta }}\]
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6.8.14.3 [1844] Problem 3
problem number 1844
Added Oct 18, 2019.
Problem Chapter 8.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta x) w_z = c \sin ^k(\gamma x) w \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*z]^m*D[w[x,y,z],z]== c*Sin[gamma*x]^k*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
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*z)^m*diff(w(x,y,z),z)= c*sin(gamma*x)^k*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-y +a \int \sin \left (\lambda x \right )^{n}d x , -\int _{}^{y}{\sin \left (\lambda \operatorname {RootOf}\left (\textit {\_b} -a \int _{}^{\textit {\_Z}}\sin \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} -y +a \int \sin \left (\lambda x \right )^{n}d x \right )\right )}^{-n}d \textit {\_b} +\frac {a \int \sin \left (\beta z \right )^{-m}d z}{b}\right ) {\mathrm e}^{\frac {c \int _{}^{y}{\sin \left (\gamma \operatorname {RootOf}\left (\textit {\_b} -a \int _{}^{\textit {\_Z}}\sin \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} -y +a \int \sin \left (\lambda x \right )^{n}d x \right )\right )}^{k} {\sin \left (\lambda \operatorname {RootOf}\left (\textit {\_b} -a \int _{}^{\textit {\_Z}}\sin \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} -y +a \int \sin \left (\lambda x \right )^{n}d x \right )\right )}^{-n}d \textit {\_b}}{a}}\]
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6.8.14.4 [1845] Problem 4
problem number 1845
Added Oct 18, 2019.
Problem Chapter 8.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta y) w_z = \left ( c \sin ^k(\gamma y) + s \sin ^r(\mu z) \right ) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*y]^m*D[w[x,y,z],z]== (c*Sin[gamma*y]^k+s*Sin[mu*z]^r)*w[x,y,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 {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 },z-\int _1^xb \sin ^m\left (\frac {\beta \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 )dK[1]\right ) \exp \left (\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[2])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda K[2])\right ) \sec (\lambda K[2]) \sin ^{n+1}(\lambda K[2])+\lambda (n+1) y\right )}{\lambda (n+1)}\right )+s \sin ^r\left (\mu \left (z-\int _1^xb \sin ^m\left (\frac {\beta \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 )dK[1]+\int _1^{K[2]}b \sin ^m\left (\frac {\beta \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 )dK[1]\right )\right )\right )dK[2]\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*y)^m*diff(w(x,y,z),z)= (c*sin(gamma*y)^k+s*sin(mu*z)^r)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \sin \left (\lambda x \right )^{n}d x +y , -b \int _{}^{x}{\sin \left (\beta \left (a \int \sin \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -a \int \sin \left (\lambda x \right )^{n}d x +y \right )\right )}^{m}d \textit {\_g} +z \right ) {\mathrm e}^{\int _{}^{x}\left (c {\sin \left (\gamma \left (a \int \sin \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -a \int \sin \left (\lambda x \right )^{n}d x +y \right )\right )}^{k}+s {\sin \left (\mu \left (-b \int _{}^{x}{\sin \left (\beta \left (a \int \sin \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -a \int \sin \left (\lambda x \right )^{n}d x +y \right )\right )}^{m}d \textit {\_g} +b \int {\sin \left (\beta \left (a \int \sin \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -a \int \sin \left (\lambda x \right )^{n}d x +y \right )\right )}^{m}d \textit {\_g} +z \right )\right )}^{r}\right )d \textit {\_g}}\]
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6.8.14.5 [1846] Problem 5
problem number 1846
Added Oct 18, 2019.
Problem Chapter 8.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin (\beta y) w_y + c \sin (\lambda x) w_z = k \sin (\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] + c*Sin[lambda*x]^m*D[w[x,y,z],z]== k*Sin[gamma*z]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\begin{align*}& \left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\beta y))}{\beta },z-\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda x)\right )}{a \lambda m+a \lambda }\right ) \exp \left (\int _1^x\frac {k \sin \left (\frac {\gamma \left (-c \sqrt {\cos ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda x)\right ) \sec (\lambda x) \sin ^{m+1}(\lambda x)+c \sqrt {\cos ^2(\lambda K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda K[1])\right ) \sec (\lambda K[1]) \sin ^{m+1}(\lambda K[1])+a \lambda (m+1) z\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\beta y))}{\beta },z-\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda x)\right )}{a \lambda m+a \lambda }\right ) \exp \left (\int _1^x\frac {k \sin \left (\frac {\gamma \left (-c \sqrt {\cos ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda x)\right ) \sec (\lambda x) \sin ^{m+1}(\lambda x)+c \sqrt {\cos ^2(\lambda K[2])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda K[2])\right ) \sec (\lambda K[2]) \sin ^{m+1}(\lambda K[2])+a \lambda (m+1) z\right )}{a \lambda (m+1)}\right )}{a}dK[2]\right )\right \}\\\end{align*}
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+b*sin(beta*y)*diff(w(x,y,z),y)+ c*sin(lambda*x)^m*diff(w(x,y,z),z)= k*sin(gamma*z)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {x b \beta +a \ln \left (\csc \left (\beta y \right )+\cot \left (\beta y \right )\right )}{b \beta }, -\frac {c \int _{}^{y}\sin \left (\frac {\lambda \left (x b \beta +a \ln \left (\csc \left (\beta y \right )+\cot \left (\beta y \right )\right )-a \ln \left (\csc \left (\beta \textit {\_b} \right )+\cot \left (\beta \textit {\_b} \right )\right )\right )}{b \beta }\right )^{m} \csc \left (\beta \textit {\_b} \right )d \textit {\_b}}{b}+z \right ) {\mathrm e}^{\frac {k \int _{}^{y}\sin \left (\frac {\gamma \left (c \int \sin \left (\frac {\lambda \left (x b \beta +a \ln \left (\csc \left (\beta y \right )+\cot \left (\beta y \right )\right )-a \ln \left (\csc \left (\beta \textit {\_b} \right )+\cot \left (\beta \textit {\_b} \right )\right )\right )}{b \beta }\right )^{m} \csc \left (\beta \textit {\_b} \right )d \textit {\_b} -c \int _{}^{y}\sin \left (\frac {\lambda \left (x b \beta +a \ln \left (\csc \left (\beta y \right )+\cot \left (\beta y \right )\right )-a \ln \left (\csc \left (\beta \textit {\_b} \right )+\cot \left (\beta \textit {\_b} \right )\right )\right )}{b \beta }\right )^{m} \csc \left (\beta \textit {\_b} \right )d \textit {\_b} +z b \right )}{b}\right ) \csc \left (\beta \textit {\_b} \right )d \textit {\_b}}{b}}\]
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6.8.14.6 [1847] Problem 6
problem number 1847
Added Oct 18, 2019.
Problem Chapter 8.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \sin ^{n_1}(\lambda _1 x) w_x + b_1 \sin ^{m_1}(\beta _1 y) w_y + c_1 \sin ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \sin ^{n_2}(\lambda _2 x) + b_2 \sin ^{m_2}(\beta _2 y) + c_2 \sin ^{k_2}(\gamma _2 z) \right ) w \]
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)*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a1*sin(lambda1*z)^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*z)^n2 + b2*sin(beta2*y)^m2 + c2*sin(gamma2*z)^k2)*w(x,y,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 (-\int \sin \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y +\frac {\operatorname {b1} \int \sin \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}, -\frac {\operatorname {a1} \int _{}^{y}{\sin \left (\lambda \operatorname {1} \operatorname {RootOf}\left (-\int \sin \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} +\operatorname {b1} \int \sin \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z -\operatorname {b1} \int _{}^{\textit {\_Z}}\sin \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} +\int \sin \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} \right )\right )}^{\operatorname {n1}} \sin \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f}}{\operatorname {b1}}+x \right ) {\mathrm e}^{\frac {\int _{}^{y}\left (\operatorname {a2} {\sin \left (\lambda \operatorname {2} \operatorname {RootOf}\left (-\int \sin \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} +\operatorname {b1} \int \sin \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z -\operatorname {b1} \int _{}^{\textit {\_Z}}\sin \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} +\int \sin \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} \right )\right )}^{\operatorname {n2}}+\operatorname {b2} \sin \left (\beta \operatorname {2} \textit {\_f} \right )^{\operatorname {m2}}+\operatorname {c2} {\sin \left (\gamma \operatorname {2} \operatorname {RootOf}\left (-\int \sin \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} +\operatorname {b1} \int \sin \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z -\operatorname {b1} \int _{}^{\textit {\_Z}}\sin \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} +\int \sin \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} \right )\right )}^{\operatorname {k2}}\right ) \sin \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f}}{\operatorname {b1}}}\]
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