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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {z a -c x}{a}\right ) {\mathrm e}^{\frac {\left (n +1\right ) \left (k +1\right ) a \beta c y^{m +1}+\left (\left (n +1\right ) a \gamma z^{k +1}+\left (k +1\right ) c \lambda x^{n +1}\right ) \left (m +1\right ) b}{\left (m +1\right ) \left (k +1\right ) \left (n +1\right ) a b c}}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\frac {b x}{a}}, z \,{\mathrm e}^{-\frac {c x}{a}}\right ) {\mathrm e}^{\frac {b c k \lambda m x^{n +1}+\left (n +1\right ) \left (b \gamma m z^{k}+\beta c k y^{m}\right ) a}{\left (n +1\right ) a b c k m}}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a z^{2}-b y^{2}}{a}, -\frac {-\sqrt {a b}\, x +\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ) {\mathrm e}^{\int _{}^{y}\frac {c \left (\frac {\sqrt {a b}\, x +\ln \left (\frac {\mathit {\_a} a b +\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )^{n}}{\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}}d\mathit {\_a}}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y x^{-\frac {b}{a}}, z x^{-\frac {c}{a}}\right ) {\mathrm e}^{\int _{}^{x}\frac {\beta \left (y \mathit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )^{m}+\gamma \left (z \mathit {\_a}^{\frac {c}{a}} x^{-\frac {c}{a}}\right )^{k}+\lambda \mathit {\_a}^{n}}{\mathit {\_a} a}d\mathit {\_a}}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a z^{2}-b y^{2}}{a}, x \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right ) {\mathrm e}^{\int _{}^{y}\frac {c \left (x \left (\frac {\mathit {\_a} a b +\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )^{\frac {1}{\sqrt {a b}}} \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right )^{n}}{\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}}d\mathit {\_a}}\]
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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]];
Failed
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\frac {1}{\sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}, x \left (\frac {\frac {\sqrt {2}\, a^{2} y}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}+\left (\frac {a y}{\sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}+\frac {b z}{\sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}\right ) \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}{\sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, a}{2 \sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}}\right ) {\mathrm e}^{-\left (\int _{}^{y}-\frac {c \left (x \left (\frac {\frac {\sqrt {2}\, \mathit {\_a} a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}+\sqrt {\frac {2 \mathit {\_a}^{2} a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}+1}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}{\sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}\right )^{\frac {\sqrt {2}\, a}{2 \sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}} \left (\frac {\frac {\sqrt {2}\, a^{2} y}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}+\left (\frac {a y}{\sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}+\frac {b z}{\sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}\right ) \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}{\sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, a}{2 \sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}}}}\right )^{n}}{\sqrt {-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}\, \sqrt {\frac {2 \mathit {\_a}^{2} a^{2}}{-a^{2} y^{2}+2 z a b y +b^{2} z^{2}}+1}\, b}d\mathit {\_a} \right )}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {\left (m -1\right ) a x^{n +1}+\left (n +1\right ) y^{-m +1}}{n +1}, \left (\lambda -1\right ) b \left (\int _{}^{x}\mathit {\_a}^{\nu } \left (\left (\frac {\left (-\mathit {\_a}^{n +1}+x^{n +1}\right ) \left (m -1\right ) a +\left (n +1\right ) y^{-m +1}}{n +1}\right )^{-\frac {1}{m -1}}\right )^{\mu }d\mathit {\_a} \right )+z^{-\lambda +1}\right ) {\mathrm e}^{\frac {c x^{k +1}}{k +1}}\]
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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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) \operatorname {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}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[2]^{\text {n1}+1}}{\text {n1}+1}\right ) \operatorname {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 {n1}+\text {n2}+2}{\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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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 {n1}+\text {n2}+2}{\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}} \operatorname {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 {n1}+\text {n2}+2}{\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}} \operatorname {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}} \operatorname {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 \operatorname {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}} \operatorname {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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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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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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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]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left (\text {b1} (-1)^{\frac {\text {n1}}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {1}{\text {n1}+1}} (k-1)^{\frac {\text {n1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {1}{\text {n1}+1},-\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}\right )+y^{1-k} e^{\frac {\text {a1} (k-1) x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (-1)^{\frac {\text {n2}}{\text {n2}+1}} (\text {n2}+1)^{-\frac {\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {1}{\text {n2}+1}} (m-1)^{\frac {\text {n2}}{\text {n2}+1}} \operatorname {Gamma}\left (\frac {1}{\text {n2}+1},-\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}\right )+z^{1-m} e^{\frac {\text {a2} (m-1) x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]
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 ) = \mathit {\_F1} \left (\frac {-\left (\mathit {n1} +1\right ) \left (\mathit {n1} +2\right )^{2} \mathit {b1} x^{-\mathit {n1}} \left (-\frac {\left (k -1\right ) \mathit {a1} x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {n1} +1\right )^{2} \left (\left (k -1\right ) \mathit {a1} x +\left (-\mathit {n1} -2\right ) x^{-\mathit {n1}}\right ) \mathit {b1} \left (-\frac {\left (k -1\right ) \mathit {a1} x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (-\frac {\mathit {n1}}{2 \mathit {n1} +2}, \frac {2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (k -1\right ) \mathit {a1} x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+2 \left (\mathit {n1} +2\right ) \left (\mathit {n1} +\frac {3}{2}\right ) \mathit {a1} y^{-k +1} {\mathrm e}^{\frac {\left (k -1\right ) \mathit {a1} x^{\mathit {n1} +1}}{\mathit {n1} +1}}}{\left (2 \mathit {n1}^{2}+7 \mathit {n1} +6\right ) \mathit {a1}}, \frac {-\left (\mathit {n2} +1\right ) \left (\mathit {n2} +2\right )^{2} \mathit {b2} x^{-\mathit {n2}} \left (-\frac {\left (m -1\right ) \mathit {a2} x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (\frac {\mathit {n2} +2}{2 \mathit {n2} +2}, \frac {2 \mathit {n2} +3}{2 \mathit {n2} +2}, -\frac {\left (m -1\right ) \mathit {a2} x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{\frac {\left (m -1\right ) \mathit {a2} x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+\left (\mathit {n2} +1\right )^{2} \left (\left (m -1\right ) \mathit {a2} x +\left (-\mathit {n2} -2\right ) x^{-\mathit {n2}}\right ) \mathit {b2} \left (-\frac {\left (m -1\right ) \mathit {a2} x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (-\frac {\mathit {n2}}{2 \mathit {n2} +2}, \frac {2 \mathit {n2} +3}{2 \mathit {n2} +2}, -\frac {\left (m -1\right ) \mathit {a2} x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{\frac {\left (m -1\right ) \mathit {a2} x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+2 \left (\mathit {n2} +2\right ) \left (\mathit {n2} +\frac {3}{2}\right ) \mathit {a2} z^{-m +1} {\mathrm e}^{\frac {\left (m -1\right ) \mathit {a2} x^{\mathit {n2} +1}}{\mathit {n2} +1}}}{\left (2 \mathit {n2}^{2}+7 \mathit {n2} +6\right ) \mathit {a2}}\right ) {\mathrm e}^{\frac {c x^{s +1}}{s +1}}\]
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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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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y}{x}, -\sqrt {x^{2}+y^{2}}\, a +z \right ) {\mathrm e}^{\frac {b x^{n}}{n}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {b x^n}{n}} c_1\left (\frac {y}{x},\log \left (-\sqrt {\frac {x^{2 a} \left (y^2+2 z^2\right )-2 \sqrt {z^2 x^{4 a} \left (x^2+y^2+z^2\right )}+x^{2 a+2}}{x^2+y^2}}\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y}{x}, \left (z +\sqrt {x^{2}+y^{2}+z^{2}}\right ) x^{a -1}\right ) {\mathrm e}^{\frac {b x^{n}}{n}}\]
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