6.8.4 2.4
6.8.4.1 [1773] Problem 1
problem number 1773
Added July 1, 2019.
Problem Chapter 8.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c w_z = (\lambda x^n + \beta y^m + \gamma z^k)w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*D[w[x,y,z],z]== (lambda*x^n+beta*y^m+gamma*z^k)*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right ) \exp \left (\frac {\lambda x^{n+1}}{a n+a}+\frac {\beta y^{m+1}}{b m+b}+\frac {\gamma z^{k+1}}{c k+c}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+c*diff(w(x,y,z),z)= (lambda*x^n+beta*y^m+gamma*z^k)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a -b x}{a}, \frac {z a -c x}{a}\right ) {\mathrm e}^{\frac {\beta \,y^{1+m}}{b \left (1+m \right )}+\frac {\gamma \,z^{1+k}}{c \left (1+k \right )}+\frac {\lambda \,x^{1+n}}{a \left (1+n \right )}}\]
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6.8.4.2 [1774] Problem 2
problem number 1774
Added July 1, 2019.
Problem Chapter 8.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b y w_y + c z w_z = (\lambda x^n + \beta y^m + \gamma z^k)w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +c*z*D[w[x,y,z],z]== (lambda*x^n+beta*y^m+gamma*z^k)*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-\frac {b x}{a}},z e^{-\frac {c x}{a}}\right ) \exp \left (\frac {\lambda x^{n+1}}{a n+a}+\frac {\beta y^m}{b m}+\frac {\gamma z^k}{c k}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+b*y*diff(w(x,y,z),y)+c*z*diff(w(x,y,z),z)= (lambda*x^n+beta*y^m+gamma*z^k)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (y \,{\mathrm e}^{-\frac {b x}{a}}, z \,{\mathrm e}^{-\frac {c x}{a}}\right ) {\mathrm e}^{\frac {\beta \,y^{m}}{m b}+\frac {\gamma \,z^{k}}{k c}+\frac {\lambda \,x^{1+n}}{a \left (1+n \right )}}\]
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6.8.4.3 [1775] Problem 3
problem number 1775
Added July 1, 2019.
Problem Chapter 8.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a z w_y + b y w_z = c x^n w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]== c*x^n*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{n+1}}{n+1}} c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+a*z*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= c*x^n*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {a \,z^{2}-y^{2} b}{a}, \frac {x \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ) {\mathrm e}^{c \int _{}^{y}\frac {{\left (\frac {x \sqrt {a b}+\ln \left (\frac {a b \textit {\_a} +\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}\, \sqrt {a b}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}^{n}}{\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}d \textit {\_a}}\]
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6.8.4.4 [1776] Problem 4
problem number 1776
Added July 1, 2019.
Problem Chapter 8.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + c z w_z = (\lambda x^n + \beta y^m + \gamma z^k) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +c*z*D[w[x,y,z],z]== (lambda*x^n+beta*y^m+gamma*z^k)*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right ) e^{\frac {\lambda x^n}{a n}+\frac {\beta y^m}{b m}+\frac {\gamma z^k}{c k}}\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*x*diff(w(x,y,z),x)+b*y*diff(w(x,y,z),y)+c*z*diff(w(x,y,z),z)= (lambda*x^n+beta*y^m+gamma*z^k)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (y \,x^{-\frac {b}{a}}, z \,x^{-\frac {c}{a}}\right ) {\mathrm e}^{\frac {\int _{}^{x}\frac {\lambda \,\textit {\_a}^{n}+\beta \left (y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )^{m}+\gamma \left (z \,x^{-\frac {c}{a}} \textit {\_a}^{\frac {c}{a}}\right )^{k}}{\textit {\_a}}d \textit {\_a}}{a}}\]
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6.8.4.5 [1777] Problem 5
problem number 1777
Added July 1, 2019.
Problem Chapter 8.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + a z w_y + b y w_z = c x^n w \]
Mathematica ✓
ClearAll["Global`*"];
pde = x*D[w[x, y,z], x] + a*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]== c*x^n*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^n}{n}} c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := x*diff(w(x,y,z),x)+a*z*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= c*x^n*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {a \,z^{2}-b \,y^{2}}{a}, x \left (\sqrt {a b}\, y +a z \right )^{-\frac {\sqrt {a b}}{a b}}\right ) {\mathrm e}^{c \int _{}^{y}\frac {\left (x \left (\sqrt {a b}\, y +a z \right )^{-\frac {\sqrt {a b}}{a b}} {\left (\frac {b \textit {\_a} a +\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}\, \sqrt {a b}}{\sqrt {a b}}\right )}^{\frac {1}{\sqrt {a b}}}\right )^{n}}{\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}d \textit {\_a}}\]
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6.8.4.6 [1778] Problem 6
problem number 1778
Added July 1, 2019.
Problem Chapter 8.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a b x w_x + b(a y + b z) w_y + a(a y - b z) w_z = c x^n w \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*b*x*D[w[x, y,z], x] + b*(a*y+b*z)*D[w[x, y,z], y] +a*(a*y-b*z)*D[w[x,y,z],z]== c*x^n*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^n}{a b n}} c_1\left (-\frac {x^{\sqrt {2}} \left (\left (\sqrt {2}-2\right ) a y+\sqrt {2} b z\right )}{4 a},\frac {x^{-\sqrt {2}} \left (\left (2+\sqrt {2}\right ) a y+\sqrt {2} b z\right )}{4 a}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*b*x*diff(w(x,y,z),x)+b*(a*y+b*z)*diff(w(x,y,z),y)+a*(a*y-b*z)*diff(w(x,y,z),z)= c*x^n*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-\frac {1}{\sqrt {-a^{2} y^{2}+2 a z b y +b^{2} z^{2}}}, x {\left (-\frac {\left (a^{3} y^{3}-a^{2} b \,y^{2} z -3 a \,b^{2} y \,z^{2}-b^{3} z^{3}\right ) \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}-a^{2} y \sqrt {-2 a^{2} y^{2}+4 a z b y +2 b^{2} z^{2}}}{\left (-a^{2} y^{2}+2 a z b y +b^{2} z^{2}\right )^{{3}/{2}} \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}}\right )}^{-\frac {a}{\sqrt {-2 a^{2} y^{2}+4 a z b y +2 b^{2} z^{2}}\, \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}}}\right ) {\mathrm e}^{\frac {c \int _{}^{y}\frac {{\left (x {\left (-\frac {\left (a^{3} y^{3}-a^{2} b \,y^{2} z -3 a \,b^{2} y \,z^{2}-b^{3} z^{3}\right ) \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}-a^{2} y \sqrt {-2 a^{2} y^{2}+4 a z b y +2 b^{2} z^{2}}}{\left (-a^{2} y^{2}+2 a z b y +b^{2} z^{2}\right )^{{3}/{2}} \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}}\right )}^{-\frac {a}{\sqrt {-2 a^{2} y^{2}+4 a z b y +2 b^{2} z^{2}}\, \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}}} {\left (\frac {\sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}\, \left (a^{2} y^{2}-2 a z b y -b^{2} z^{2}\right ) \sqrt {\frac {\left (-2 \textit {\_a}^{2}+y^{2}\right ) a^{2}-2 a z b y -b^{2} z^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}-a^{2} \textit {\_a} \sqrt {2}}{\sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}\, \left (a^{2} y^{2}-2 a z b y -b^{2} z^{2}\right )}\right )}^{\frac {a}{\sqrt {-2 a^{2} y^{2}+4 a z b y +2 b^{2} z^{2}}\, \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}}}\right )}^{n}}{\sqrt {\frac {\left (-2 \textit {\_a}^{2}+y^{2}\right ) a^{2}-2 a z b y -b^{2} z^{2}}{a^{2} y^{2}-2 a z b y -b^{2} z^{2}}}}d \textit {\_a}}{\sqrt {-a^{2} y^{2}+2 a z b y +b^{2} z^{2}}\, b}}\]
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6.8.4.7 [1779] Problem 7
problem number 1779
Added July 1, 2019.
Problem Chapter 8.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a x^n y^m w_y + b x^\nu y^\mu z^\lambda w_z = c x^k w \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*x^n*y^m*D[w[x, y,z], y] +b*x^nu*y^mu*z^lambda*D[w[x,y,z],z]== c*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*x^n*y^m*diff(w(x,y,z),y)+b*x^nu*y^mu*z^lambda*diff(w(x,y,z),z)= c*x^k*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (n +1\right ) y^{1-m}+x^{n +1} \left (m -1\right ) a}{n +1}, \textit {\_a}^{\nu } b x \left (\lambda -1\right ) {\left (\left (\frac {\left (n +1\right ) y^{1-m}+a \left (-\textit {\_a}^{n +1}+x^{n +1}\right ) \left (m -1\right )}{n +1}\right )^{-\frac {1}{m -1}}\right )}^{\mu }+z^{1-\lambda }\right ) {\mathrm e}^{\frac {c \,x^{k +1}}{k +1}}\]
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6.8.4.8 [1780] Problem 8
problem number 1780
Added July 1, 2019.
Problem Chapter 8.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^{n_1} y +b_1 x^{m_1} ) w_y + (a_2 x^{n_2} y +b_2 x^{m_2} ) w_z = (c_1 x^{k_2} y +c_1 x^{k_1} z ) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a1*x^n1*y+b1*x^m1)*D[w[x, y,z], y] +(a2*x^n2*y+b2*x^m2)*D[w[x,y,z],z]== (c1*x^k2*y+c1*x^k1*z)*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^{\int _1^x\frac {(-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} \text {c1} e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \left ((-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} z K[2]^{\text {k1}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \text {n1}^2 z K[2]^{\text {k1}}+2 (-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \text {n1} z K[2]^{\text {k1}}+\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}+\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} \text {n1} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}-\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}-\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} \text {a2} \text {n1} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} y \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}-\text {a2} \text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}+\text {a2} \text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k1}}-(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^2 \int _1^x\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\text {n1}+1}} \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right ) K[1]^{\text {m1}}+\text {b2} K[1]^{\text {m2}}\right )dK[1] K[2]^{\text {k1}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^2 \int _1^{K[2]}\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\text {n1}+1}} \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right ) K[1]^{\text {m1}}+\text {b2} K[1]^{\text {m2}}\right )dK[1] K[2]^{\text {k1}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}} y K[2]^{\text {k2}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}} \text {n1}^2 y K[2]^{\text {k2}}+2 (-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {m1}+\text {n2}+2}{\text {n1}+1}} e^{\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}} \text {n1} y K[2]^{\text {k2}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}+(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} \text {n1} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}-(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}-(-1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{\frac {\text {n2}+1}{\text {n1}+1}} \text {b1} e^{\frac {\text {a1} \left (x^{\text {n1}+1}+K[2]^{\text {n1}+1}\right )}{\text {n1}+1}} \text {n1} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) K[2]^{\text {k2}}\right )}{(\text {n1}+1)^2}dK[2]} c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}}+e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} y,(-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\text {n1}+1}} y \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}}+z-\int _1^x\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\text {n1}+1}} \Gamma \left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right ) K[1]^{\text {m1}}+\text {b2} K[1]^{\text {m2}}\right )dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1)*diff(w(x,y,z),y)+(a2*x^n2*y+b2*x^m2)*diff(w(x,y,z),z)= (c1*x^k2*y+c1*x^k1*z)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[\text {Expression too large to display}\]
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6.8.4.9 [1781] Problem 9
problem number 1781
Added July 1, 2019.
Problem Chapter 8.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^{n_1} y +b_1 x^{m_1} ) w_y + (a_2 x^{n_2} z +b_2 x^{m_2} ) w_z = (c_1 x^{k_2} y +c_1 x^{k_1} z) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a1*x^n1*y+b1*x^m1)*D[w[x, y,z], y] +(a2*x^n2*z+b2*x^m2)*D[w[x,y,z],z]== (c1*x^k2*y+c1*x^k1*z)*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 (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (\text {n2}+1)^{\frac {\text {m2}-\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {\text {m2}+1}{\text {n2}+1}} \Gamma \left (\frac {\text {m2}+1}{\text {n2}+1},\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}\right )+z e^{-\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}}\right ) \exp \left (\int _1^x\left (\frac {\text {a2}^{-\frac {\text {m2}+1}{\text {n2}+1}} \text {c1} e^{\frac {\text {a2} \left (K[1]^{\text {n2}+1}-x^{\text {n2}+1}\right )}{\text {n2}+1}} \left ((\text {n2}+1) z \text {a2}^{\frac {\text {m2}+1}{\text {n2}+1}}+\text {b2} e^{\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}} (\text {n2}+1)^{\frac {\text {m2}+1}{\text {n2}+1}} \Gamma \left (\frac {\text {m2}+1}{\text {n2}+1},\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}\right )-\text {b2} e^{\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}} (\text {n2}+1)^{\frac {\text {m2}+1}{\text {n2}+1}} \Gamma \left (\frac {\text {m2}+1}{\text {n2}+1},\frac {\text {a2} K[1]^{\text {n2}+1}}{\text {n2}+1}\right )\right ) K[1]^{\text {k1}}}{\text {n2}+1}+\frac {\text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \text {c1} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} \left ((\text {n1}+1) y \text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}}+\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )-\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}+1}{\text {n1}+1}} \Gamma \left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) K[1]^{\text {k2}}}{\text {n1}+1}\right )dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*x^m2)*diff(w(x,y,z),z)= (c1*x^k2*y+c1*x^k1*z)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[\text {Expression too large to display}\]
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6.8.4.10 [1782] Problem 10
problem number 1782
Added July 1, 2019.
Problem Chapter 8.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^{n_1} y +b_1 y^k ) w_y + (a_2 x^{n_2} z +b_2 z^m ) w_z = c x^s w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a1*x^n1*y+b1*y^k)*D[w[x, y,z], y] +(a2*x^n2*z+b2*z^m)*D[w[x,y,z],z]== c*x^s*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left (y^{1-k} e^{\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}}-\text {b1} (-1)^{-\frac {1}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {1}{\text {n1}+1}} (k-1)^{\frac {\text {n1}}{\text {n1}+1}} \Gamma \left (\frac {1}{\text {n1}+1},-\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}\right ),z^{1-m} e^{\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}}-\text {b2} (-1)^{-\frac {1}{\text {n2}+1}} (\text {n2}+1)^{-\frac {\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {1}{\text {n2}+1}} (m-1)^{\frac {\text {n2}}{\text {n2}+1}} \Gamma \left (\frac {1}{\text {n2}+1},-\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}\right )\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(a1*x^n1*y+b1*y^k)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*z^m)*diff(w(x,y,z),z)= c*x^s*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 {\left (-\frac {x \operatorname {a1} \,x^{\operatorname {n1}} \left (k -1\right )}{\operatorname {n1} +1}\right )^{\frac {-\operatorname {n1} -2}{2 \operatorname {n1} +2}} x^{-\operatorname {n1}} \left (-{\mathrm e}^{\frac {x^{\operatorname {n1}} x \operatorname {a1} \left (k -1\right )}{2 \operatorname {n1} +2}} \operatorname {b1} \left (\operatorname {n1} +1\right ) \left (\operatorname {n1} +2\right )^{2} \operatorname {WhittakerM}\left (\frac {\operatorname {n1} +2}{2 \operatorname {n1} +2}, \frac {2 \operatorname {n1} +3}{2 \operatorname {n1} +2}, -\frac {x \operatorname {a1} \,x^{\operatorname {n1}} \left (k -1\right )}{\operatorname {n1} +1}\right )+\left (x \operatorname {a1} \,x^{\operatorname {n1}} \left (k -1\right )-\operatorname {n1} -2\right ) \operatorname {b1} \,{\mathrm e}^{\frac {x^{\operatorname {n1}} x \operatorname {a1} \left (k -1\right )}{2 \operatorname {n1} +2}} \left (\operatorname {n1} +1\right )^{2} \operatorname {WhittakerM}\left (-\frac {\operatorname {n1}}{2 \operatorname {n1} +2}, \frac {2 \operatorname {n1} +3}{2 \operatorname {n1} +2}, -\frac {x \operatorname {a1} \,x^{\operatorname {n1}} \left (k -1\right )}{\operatorname {n1} +1}\right )+2 y \,y^{-k} \left (\operatorname {n1} +\frac {3}{2}\right ) \left (\operatorname {n1} +2\right ) x^{\operatorname {n1}} {\mathrm e}^{\frac {x \operatorname {a1} \,x^{\operatorname {n1}} \left (k -1\right )}{\operatorname {n1} +1}} \left (-\frac {x \operatorname {a1} \,x^{\operatorname {n1}} \left (k -1\right )}{\operatorname {n1} +1}\right )^{\frac {\operatorname {n1} +2}{2 \operatorname {n1} +2}} \operatorname {a1} \right )}{\operatorname {a1} \left (2 \operatorname {n1}^{2}+7 \operatorname {n1} +6\right )}, \frac {\left (-\frac {x \operatorname {a2} \,x^{\operatorname {n2}} \left (m -1\right )}{\operatorname {n2} +1}\right )^{\frac {-\operatorname {n2} -2}{2 \operatorname {n2} +2}} x^{-\operatorname {n2}} \left (-{\mathrm e}^{\frac {x^{\operatorname {n2}} x \operatorname {a2} \left (m -1\right )}{2 \operatorname {n2} +2}} \operatorname {b2} \left (\operatorname {n2} +1\right ) \left (\operatorname {n2} +2\right )^{2} \operatorname {WhittakerM}\left (\frac {\operatorname {n2} +2}{2 \operatorname {n2} +2}, \frac {2 \operatorname {n2} +3}{2 \operatorname {n2} +2}, -\frac {x \operatorname {a2} \,x^{\operatorname {n2}} \left (m -1\right )}{\operatorname {n2} +1}\right )+\left (x \operatorname {a2} \,x^{\operatorname {n2}} \left (m -1\right )-\operatorname {n2} -2\right ) \operatorname {b2} \,{\mathrm e}^{\frac {x^{\operatorname {n2}} x \operatorname {a2} \left (m -1\right )}{2 \operatorname {n2} +2}} \left (\operatorname {n2} +1\right )^{2} \operatorname {WhittakerM}\left (-\frac {\operatorname {n2}}{2 \operatorname {n2} +2}, \frac {2 \operatorname {n2} +3}{2 \operatorname {n2} +2}, -\frac {x \operatorname {a2} \,x^{\operatorname {n2}} \left (m -1\right )}{\operatorname {n2} +1}\right )+2 z \,z^{-m} \left (\operatorname {n2} +\frac {3}{2}\right ) \left (\operatorname {n2} +2\right ) x^{\operatorname {n2}} {\mathrm e}^{\frac {x \operatorname {a2} \,x^{\operatorname {n2}} \left (m -1\right )}{\operatorname {n2} +1}} \left (-\frac {x \operatorname {a2} \,x^{\operatorname {n2}} \left (m -1\right )}{\operatorname {n2} +1}\right )^{\frac {\operatorname {n2} +2}{2 \operatorname {n2} +2}} \operatorname {a2} \right )}{\operatorname {a2} \left (2 \operatorname {n2}^{2}+7 \operatorname {n2} +6\right )}\right ) {\mathrm e}^{\frac {x c \,x^{s}}{s +1}}\]
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6.8.4.11 [1783] Problem 11
problem number 1783
Added July 1, 2019.
Problem Chapter 8.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^{n_1} y +b_1 y^k ) w_y + (a_2 y^{n_2} z +b_2 z^m ) w_z = (c_1 x^{s_1} + c_2 y^{s_2} + c_3 z^{s_3}) w \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a1*x^n1*y+b1*y^k)*D[w[x, y,z], y] +(a2*y^n2*z+b2*z^m)*D[w[x,y,z],z]== (c1*x^s1+c2*y^s2+c3*z^s3)*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)+(a1*x^n1*y+b1*y^k)*diff(w(x,y,z),y)+(a2*y^n2*z+b2*z^m)*diff(w(x,y,z),z)= (c1*x^s1+c2*y^s2+c3*z^s3)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[\text {Expression too large to display}\]
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6.8.4.12 [1784] Problem 12
problem number 1784
Added July 1, 2019.
Problem Chapter 8.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + a \sqrt {x^2+y^2} w_z = b x^n w \]
Mathematica ✓
ClearAll["Global`*"];
pde = x*D[w[x, y,z], x] + y*D[w[x, y,z], y] +a*Sqrt[x^2+y^2]*D[w[x,y,z],z]== b*x^n*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {b x^n}{n}} c_1\left (\frac {y}{x},z-a \sqrt {x^2+y^2}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := x*diff(w(x,y,z),x)+y*diff(w(x,y,z),y)+a*sqrt(x^2+y^2)*diff(w(x,y,z),z)= b*x^n*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, -a \sqrt {x^{2}+y^{2}}+z \right ) {\mathrm e}^{\frac {b \,x^{n}}{n}}\]
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6.8.4.13 [1785] Problem 13
problem number 1785
Added July 1, 2019.
Problem Chapter 8.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + (z - a \sqrt {x^2+y^2+z^2} w_z = b x^n w \]
Mathematica ✓
ClearAll["Global`*"];
pde = x*D[w[x, y,z], x] + y*D[w[x, y,z], y] +(z-a*Sqrt[x^2+y^2+z^2])*D[w[x,y,z],z]== b*x^n*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {b x^n}{n}} c_1\left (\frac {y}{x},\log \left (\frac {z x^a-x \sqrt {x^{2 a-2} \left (x^2+y^2+z^2\right )}}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := x*diff(w(x,y,z),x)+y*diff(w(x,y,z),y)+(z-a*sqrt(x^2+y^2+z^2))*diff(w(x,y,z),z)= b*x^n*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, x^{a -1} \left (z +\sqrt {x^{2}+y^{2}+z^{2}}\right )\right ) {\mathrm e}^{\frac {b \,x^{n}}{n}}\]
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