6.8.3 2.3

6.8.3.1 [1768] Problem 1

problem number 1768

Added July 1, 2019.

Problem Chapter 8.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a w_y + b w_z = x y z w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*D[w[x, y,z], y] +b*D[w[x,y,z],z]== x*y*z*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {1}{12} x^2 (a x (b x-2 z)-2 b x y+6 y z)\right ) c_1(y-a x,z-b x)\right \}\right \}\]

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)=  x*y*z*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\frac {x^{2} \left (a b \,x^{2}-2 a x z -2 b x y +6 y z \right )}{12}}\]

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6.8.3.2 [1769] Problem 2

problem number 1769

Added July 1, 2019.

Problem Chapter 8.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c w_z = (k x^3+ s y^2) w \]

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]== (k*x^3+s*y^2)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {x \left (3 a^2 \left (k x^3+4 s y^2\right )-12 a b s x y+4 b^2 s x^2\right )}{12 a^3}\right ) c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )\right \}\right \}\]

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)=  (k*x^3+s*y^2)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a -b x}{a}, \frac {z a -c x}{a}\right ) {\mathrm e}^{\frac {x \left (3 k \,x^{3} a^{2}+12 a^{2} s \,y^{2}-12 a b s x y +4 b^{2} s \,x^{2}\right )}{12 a^{3}}}\]

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6.8.3.3 [1770] Problem 3

problem number 1770

Added July 1, 2019.

Problem Chapter 8.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b y w_y + c z w_z = (k x+ s \sqrt x) w \]

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]== (k*x+s*Sqrt[x])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {3 k x^2+4 s x^{3/2}}{6 a}} c_1\left (y e^{-\frac {b x}{a}},z e^{-\frac {c x}{a}}\right )\right \}\right \}\]

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)=  (k*x+s*sqrt(x))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (y \,{\mathrm e}^{-\frac {b x}{a}}, z \,{\mathrm e}^{-\frac {c x}{a}}\right ) {\mathrm e}^{\frac {4 s \,x^{{3}/{2}}+3 k \,x^{2}}{6 a}}\]

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6.8.3.4 [1771] Problem 4

problem number 1771

Added July 1, 2019.

Problem Chapter 8.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a z w_y + b y w_z = (c \sqrt x + s) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + b*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]== (c*Sqrt[x]+s)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {2}{3} c x^{3/2}+s x} c_1\left (\frac {1}{2} e^{-b x} \left (y \left (-e^{2 b x}\right )+z e^{2 b x}+y+z\right ),\frac {1}{2} e^{-b x} \left (y e^{2 b x}-z e^{2 b x}+y+z\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=   diff(w(x,y,z),x)+b*z*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)=  (c*sqrt(x)+s)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-y^{2}+z^{2}, \frac {x b -\ln \left (y +z \right )}{b}\right ) \left (y +z \right )^{\frac {s}{b}} {\mathrm e}^{\frac {2 c \,x^{{3}/{2}}}{3}}\]

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6.8.3.5 [1772] Problem 5

problem number 1772

Added July 1, 2019.

Problem Chapter 8.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a x^2 w_x + b y^2 w_y + c z^2 w_z = k x y z w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^2*D[w[x, y,z], x] + b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== k*x*y*z*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right ) \exp \left (\frac {k x y z \left (b y (a x-c z) \log \left (\frac {a x}{y}\right )+c z (b y-a x) \log \left (\frac {a x}{z}\right )\right )}{(a x-b y) (a x-c z) (b y-c z)}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=   a*x^2*diff(w(x,y,z),x)+b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)=  k*x*y*z*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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