Added April 15, 2019.
Problem Chapter 6.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ 2 b^2 x z w_x + b y (b^2 z^2 +1 ) w_y + a x y (b z +1)^2 w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = 2*b^2*x*z*D[w[x, y,z], x] + b*y*(b^2*z^2 +1)*D[w[x, y,z], y] + a*x*y*(b*z +1)^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✗
restart; pde := 2*b^2*x*z*diff(w(x,y,z),x)+b*y*(b^2*z^2 +1)*diff(w(x,y,z),y)+a*x*y*(b*z +1)^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 15, 2019.
Problem Chapter 6.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c x y^2 w_x + 2 b c y^3 w_y + 2(c y z-a x^2)^2 w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*c*x*y^2*D[w[x, y,z], x] + 2*b*c*y^3*D[w[x, y,z], y] + 2*(c*y*z-a*x^2)^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x^2},\frac {x^4 \left (\log (x) \left (2 c y z-2 a x^2\right )+b y\right )}{b c y^2 \left (a x^2-c y z\right )}\right )\right \}\right \}\]
Maple ✓
restart; pde := b*c*x*y^2*diff(w(x,y,z),x)+2*b*c*y^3*diff(w(x,y,z),y)+2*(c*y*z-a*x^2)^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y}{x^{2}}, \frac {b y +\left (-2 a x^{2}+2 c y z \right ) \ln \left (x \right )}{2 a x^{2}-2 c y z}\right )\]
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Added April 15, 2019.
Problem Chapter 6.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c^2 y^2 z w_x + a c^2 x z^2 w_y - a b x y^2 w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*c*x*y^2*D[w[x, y,z], x] + a*c^2*x*z^2*D[w[x, y,z], y] - a*b*x*y^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a x}{c}+z,\frac {b y^3+c^2 z^3}{3 b}\right )\right \}\right \}\]
Maple ✓
restart; pde := b*c*x*y^2*diff(w(x,y,z),x)+a*c^2*x*z^2*diff(w(x,y,z),y)- a*b*x*y^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a x +c z}{c}, \frac {-a^{3} x^{3}-3 a^{2} c x^{2} z -3 a c^{2} x z^{2}+b c y^{3}}{b c}\right )\]
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Added April 15, 2019.
Problem Chapter 6.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x(b y^2-c z^2) w_x + y(c z^2-a x^2) w_y + z(a x^2-b y^2) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*(b*y^2-c*z^2)*D[w[x, y,z], x] + y*(c*z^2-a*x^2)*D[w[x, y,z], y] + z*(a*x^2-b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde :=x*(b*y^2-c*z^2)*diff(w(x,y,z),x)+ y*(c*z^2-a*x^2)*diff(w(x,y,z),y) + z*(a*x^2-b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right ) = c_{3} c_{4} c_{5} x^{\frac {c_{2}}{2}} y^{\frac {c_{2}}{2}} z^{\frac {c_{2}}{2}} {\mathrm e}^{\frac {c_{2}}{4}} {\mathrm e}^{-\frac {c_{1} x^{2}}{4}} {\mathrm e}^{-\frac {c_{1} b y^{2}}{4 a}} {\mathrm e}^{-\frac {c_{1} c z^{2}}{4 a}}\]
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Added April 15, 2019.
Problem Chapter 6.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b y (3 a x^2+ b y^2+c z^2) w_x - 2 a x(a x^2+c z^2) w_y + 2 a b x y z w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = b*y*(3*a*x^2+ b*y^2+c*z^2)*D[w[x, y,z], x] - 2*a*x*(a*x^2+c*z^2)*D[w[x, y,z], y] + 2*a*b*x*y*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := b*y*(3*a*x^2+ b*y^2+c*z^2)*diff(w(x,y,z),x)- 2*a*x*(a*x^2+c*z^2)*diff(w(x,y,z),y) + 2*a*b*x*y*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 15, 2019.
Problem Chapter 6.2.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b(a(a^2 x^2+b^2 y^2-1) x+ b y ) w_x + a(b(a^2 x^2+b^2 y^2-1)y - a x) w_y + 2 a b z w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = b*(a*(a^2*x^2+b^2*y^2-1)*x+ b*y )*D[w[x, y,z], x] +a*(b*(a^2*x^2+b^2*y^2-1)*y - a*x)*D[w[x, y,z], y] + 2*a*b*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := b*(a*(a^2*x^2+b^2*y^2-1)*x+ b*y )*diff(w(x,y,z),x)+a*(b*(a^2*x^2+b^2*y^2-1)*y - a*x)*diff(w(x,y,z),y) + 2*a*b*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {\left (a^{2} x^{2}+b^{2} y^{2}-1\right ) {\mathrm e}^{2 \arctan \left (\frac {b y}{a x}\right )}}{a^{2} x^{2}+b^{2} y^{2}}, z \,{\mathrm e}^{2 \left (\int _{}^{x}\frac {\cos ^{2}\left (\RootOf \left (2 \mathit {\_Z} -\ln \left (-\frac {\left (a^{2} x^{2}+b^{2} y^{2}-1\right ) \mathit {\_a}^{2} a^{2} {\mathrm e}^{2 \arctan \left (\frac {b y}{a x}\right )}}{\left (a^{2} x^{2}+b^{2} y^{2}\right ) \left (-\mathit {\_a}^{2} a^{2}+\cos ^{2}\mathit {\_Z} \right )}\right )\right )\right )}{\left (-\mathit {\_a}^{2} a^{2}+\cos ^{2}\left (\RootOf \left (2 \mathit {\_Z} -\ln \left (-\frac {\left (a^{2} x^{2}+b^{2} y^{2}-1\right ) \mathit {\_a}^{2} a^{2} {\mathrm e}^{2 \arctan \left (\frac {b y}{a x}\right )}}{\left (a^{2} x^{2}+b^{2} y^{2}\right ) \left (-\mathit {\_a}^{2} a^{2}+\cos ^{2}\mathit {\_Z} \right )}\right )\right )\right )-\cos \left (\RootOf \left (2 \mathit {\_Z} -\ln \left (-\frac {\left (a^{2} x^{2}+b^{2} y^{2}-1\right ) \mathit {\_a}^{2} a^{2} {\mathrm e}^{2 \arctan \left (\frac {b y}{a x}\right )}}{\left (a^{2} x^{2}+b^{2} y^{2}\right ) \left (-\mathit {\_a}^{2} a^{2}+\cos ^{2}\mathit {\_Z} \right )}\right )\right )\right ) \sin \left (\RootOf \left (2 \mathit {\_Z} -\ln \left (-\frac {\left (a^{2} x^{2}+b^{2} y^{2}-1\right ) \mathit {\_a}^{2} a^{2} {\mathrm e}^{2 \arctan \left (\frac {b y}{a x}\right )}}{\left (a^{2} x^{2}+b^{2} y^{2}\right ) \left (-\mathit {\_a}^{2} a^{2}+\cos ^{2}\mathit {\_Z} \right )}\right )\right )\right )\right ) \mathit {\_a}}d\mathit {\_a} \right )}\right )\]
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Added April 15, 2019.
Problem Chapter 6.2.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x(b^3 y^3 - 2 a^3 x^3) w_x + y(2 b^3 y^3 -a^3 x^3) w_y + 9 z(a^3 x^3-b^3 y^3) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*(b^3*y^3 - 2*a^3*x^3)*D[w[x, y,z], x] +y*(2*b^3*y^3 -a^3*x^3)*D[w[x, y,z], y] +9*z*(a^3*x^3-b^3*y^3)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; pde := x*(b^3*y^3 - 2*a^3*x^3)*diff(w(x,y,z),x)+y*(2*b^3*y^3 -a^3*x^3)*diff(w(x,y,z),y) + 9*z*(a^3*x^3-b^3*y^3)*diff(w(x,y,z),z)= 0; 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 April 15, 2019.
Problem Chapter 6.2.3.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2(a b x y-c^2 z^2) w_x + a x y(a b x y-c^2 z^2) w_y + b y z(b c y z+2 a^2 x^2) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*(a*b*x*y-c^2*z^2)*D[w[x, y,z], x] +a*x*y*(a*b*x*y-c^2*z^2)*D[w[x, y,z], y] +b*y*z*(b*c*y*z+2*a^2*x^2) *D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {\log \left (\frac {x z}{a^2 b x^2 y+a c^2 x z^2+b^2 c y^2 z}\right )}{a}\right )\right \}\right \}\]
Maple ✗
restart; pde := a*x^2*(a*b*x*y-c^2*z^2)*diff(w(x,y,z),x)+a*x*y*(a*b*x*y-c^2*z^2)*diff(w(x,y,z),y) + b*y*z*(b*c*y*z+2*a^2*x^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 15, 2019.
Problem Chapter 6.2.3.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x(c z^4 - b y^4) w_x + y(a x^4-2 c z^4) w_y + z(2 b y^4-a x^4) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*(c*z^4 - b*y^4)*D[w[x, y,z], x] +y*(a*x^4-2*c*z^4)*D[w[x, y,z], y] +z*(2*b*y^4-a*x^4) *D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := x*(c*z^4 - b*y^4)*diff(w(x,y,z),x)+y*(a*x^4-2*c*z^4)*diff(w(x,y,z),y) + z*(2*b*y^4-a*x^4)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right ) = c_{3} c_{4} c_{5} x^{\frac {c_{2}}{4}} y^{\frac {c_{2}}{8}} z^{\frac {c_{2}}{8}} {\mathrm e}^{\frac {c_{2}}{16}} {\mathrm e}^{-\frac {c_{1} x^{4}}{16}} {\mathrm e}^{-\frac {c_{1} b y^{4}}{16 a}} {\mathrm e}^{-\frac {c_{1} c z^{4}}{16 a}}\]
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Added April 15, 2019.
Problem Chapter 6.2.3.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + a \sqrt {x^2+y^2}w_z= 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},z-a \sqrt {x^2+y^2}\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y}{x}, -\sqrt {x^{2}+y^{2}}\, a +z \right )\]
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Added April 15, 2019.
Problem Chapter 6.2.3.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + (z- a \sqrt {x^2+y^2+z^2})w_z= 0 \]
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]==0; 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 {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 \}\\& \left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {1}{2} \log \left (\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 \}\\& \left \{w(x,y,z)\to 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 \}\\& \left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {1}{2} \log \left (\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 \}\\ \end {align*}
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y}{x}, \left (z +\sqrt {x^{2}+y^{2}+z^{2}}\right ) x^{a -1}\right )\]
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Added April 15, 2019.
Problem Chapter 6.2.3.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ z \sqrt {y^2+z^2} w_x + a z \sqrt {x^2+z^2} w_y - (x \sqrt {y^2+z^2}+a y \sqrt {x^2+z^2})w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = z*Sqrt[y^2+z^2]*D[w[x, y,z], x] +a*z*Sqrt[x^2+z^2]*D[w[x, y,z], y] -(x*Sqrt[y^2+z^2]+a*y*Sqrt[x^2+z^2])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := z*sqrt(y^2+z^2)*diff(w(x,y,z),x)+a*z*sqrt(x^2+z^2)*diff(w(x,y,z),y) -(x*sqrt(y^2+z^2)+a*y*sqrt(x^2+z^2))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 15, 2019.
Problem Chapter 6.2.3.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (y-z) \sqrt {f(x)} w_x + (z-x)\sqrt {f(y)} w_y + (x-y)\sqrt {f(z)}w_z= 0 \] Where \[ f(t) = a_6 t^6 + a_5 t^5 + a_4 t^4 + a_3 t^3+ a_2 t^2 + a_1 t+a_0 \]
Mathematica ✗
ClearAll["Global`*"]; f[t_]:= a[6]*t^6+a[5]*t^5+a[4]*t^4+a[3]*t^3+a[2]*t^2+a[1]*t+a[0]; pde = (y-z)*Sqrt[f[x]]*D[w[x, y,z], x] +(z-x)*Sqrt[f[y]]*D[w[x, y,z], y] +(x-y)*Sqrt[f[z]]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; f := t-> a[6]*t^6+a[5]*t^5+a[4]*t^4+a[3]*t^3+a[2]*t^2+a[1]*t+a[0]; pde := (y-z)*sqrt(f(x))*diff(w(x,y,z),x)+(z-x)*sqrt(f(y))*diff(w(x,y,z),y)+(x-y)*sqrt(f(z))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
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