6.6.3 2.3

6.6.3.1 [1417] Problem 1

problem number 1417

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]];
 

Failed

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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6.6.3.2 [1418] Problem 2

problem number 1418

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x^{2}}, \frac {\left (-2 a \,x^{2}+2 c y z \right ) \ln \left (x \right )+y b}{2 a \,x^{2}-2 c y z}\right )\]

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6.6.3.3 [1419] Problem 3

problem number 1419

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a x +z c}{c}, \frac {c^{2} z^{3}+b \,y^{3}}{b}\right )\]

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6.6.3.4 [1420] Problem 4

problem number 1420

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'));
 

\[w \left (x , y , z\right ) = c_{3} c_{4} c_{5} x^{\frac {c_{2}}{2}} y^{\frac {c_{2}}{2}} z^{\frac {c_{2}}{2}} {\mathrm e}^{\frac {c_{2}}{4}-\frac {c_{1} x^{2}}{4}-\frac {y^{2} c_{1} b}{4 a}-\frac {z^{2} c_{1} c}{4 a}}\]

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6.6.3.5 [1421] Problem 5

problem number 1421

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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6.6.3.6 [1422] Problem 6

problem number 1422

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\frac {b \left (\ln \left (\frac {b^{2} \left (a^{2} x^{2}+b^{2} y^{2}-1\right )}{a^{2} x^{2}+b^{2} y^{2}}\right )-2 \,\operatorname {arccot}\left (\frac {y b}{a x}\right )\right )}{2 a}, z \,{\mathrm e}^{-2 \int _{}^{x}\frac {1}{\textit {\_a} \left ({\cot \left (\operatorname {RootOf}\left (b \left (\ln \left (\frac {b^{2} \left (a^{2} x^{2}+b^{2} y^{2}-1\right )}{a^{2} x^{2}+b^{2} y^{2}}\right )-2 \,\operatorname {arccot}\left (\frac {y b}{a x}\right )-\ln \left (\frac {b^{2} \left (2 a^{2} \textit {\_a}^{2}+\cos \left (2 \textit {\_Z} \right )-1\right )}{a^{2} \textit {\_a}^{2}}\right )+\ln \left (2\right )+2 \textit {\_Z} \right )\right )\right )}^{2} a^{2} \textit {\_a}^{2}+a^{2} \textit {\_a}^{2}+\cot \left (\operatorname {RootOf}\left (b \left (\ln \left (\frac {b^{2} \left (a^{2} x^{2}+b^{2} y^{2}-1\right )}{a^{2} x^{2}+b^{2} y^{2}}\right )-2 \,\operatorname {arccot}\left (\frac {y b}{a x}\right )-\ln \left (\frac {b^{2} \left (2 a^{2} \textit {\_a}^{2}+\cos \left (2 \textit {\_Z} \right )-1\right )}{a^{2} \textit {\_a}^{2}}\right )+\ln \left (2\right )+2 \textit {\_Z} \right )\right )\right )-1\right )}d \textit {\_a}}\right )\]

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6.6.3.7 [1423] Problem 7

problem number 1423

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]];
 

Failed

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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6.6.3.8 [1424] Problem 8

problem number 1424

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]];
 

Failed

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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6.6.3.9 [1425] Problem 9

problem number 1425

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'));
 

\[w \left (x , y , z\right ) = c_{3} c_{4} c_{5} x^{\frac {c_{2}}{4}} y^{\frac {c_{2}}{8}} z^{\frac {c_{2}}{8}} {\mathrm e}^{\frac {\left (-x^{4} a -b \,y^{4}-z^{4} c \right ) c_{1} +c_{2} a}{16 a}}\]

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6.6.3.10 [1426] Problem 10

problem number 1426

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, -a \sqrt {x^{2}+y^{2}}+z \right )\]

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6.6.3.11 [1427] Problem 11

problem number 1427

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 (\frac {z x^a-x \sqrt {x^{2 a-2} \left (x^2+y^2+z^2\right )}}{x}\right )\right )\right \}\\& \left \{w(x,y,z)\to 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 \}\\\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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, x^{a -1} \left (z +\sqrt {x^{2}+y^{2}+z^{2}}\right )\right )\]

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6.6.3.12 [1428] Problem 12

problem number 1428

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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6.6.3.13 [1429] Problem 13

problem number 1429

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'));
 

detected error

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