6.7.3 2.3

6.7.3.1 [1597] Problem 1

problem number 1597

Added June 1, 2019.

Problem Chapter 7.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-xb+z \right ) +{\frac {ab{x}^{4}}{12}}+{\frac { \left ( -2\,za-2\,by \right ) {x}^{3}}{12}}+{\frac {{x}^{2}yz}{2}}\]

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6.7.3.2 [1598] Problem 2

problem number 1598

Added June 1, 2019.

Problem Chapter 7.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) ={\frac {k{x}^{4}}{4\,a}}+{\frac {{b}^{2}s{x}^{3}}{3\,{a}^{3}}}-{\frac {bsy{x}^{2}}{{a}^{2}}}+{\frac {s{y}^{2}x}{a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {za-cx}{a}} \right ) \]

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6.7.3.3 [1599] Problem 3

problem number 1599

Added June 1, 2019.

Problem Chapter 7.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) ={\frac {k{x}^{2}}{2\,a}}+{\frac {2\,s}{3\,a}{x}^{{\frac {3}{2}}}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {xb}{a}}}},z{{\rm e}^{-{\frac {cx}{a}}}} \right ) \]

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6.7.3.4 [1600] Problem 4

problem number 1600

Added June 1, 2019.

Problem Chapter 7.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

\[w \left ( x,y,z \right ) =\int ^{y}\!{ \left ( c\sqrt {{ \left ( x\sqrt {ba}-\ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) +\ln \left ( { \left ( {\it \_a}\,ab+\sqrt { \left ( {z}^{2}a+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) a}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) \right ) {\frac {1}{\sqrt {ba}}}}}+s \right ) {\frac {1}{\sqrt { \left ( {z}^{2}a+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) a}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {{z}^{2}a-b{y}^{2}}{a}},-{ \left ( -x\sqrt {ba}+\ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) \right ) {\frac {1}{\sqrt {ba}}}} \right ) \]

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6.7.3.5 [1601] Problem 5

problem number 1601

Added June 1, 2019.

Problem Chapter 7.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) ={\frac {1}{ \left ( by-cz \right ) \left ( ax-cz \right ) \left ( ax-by \right ) } \left ( \left ( by-cz \right ) \left ( ax-cz \right ) \left ( ax-by \right ) {\it \_F1} \left ( {\frac {ax-by}{ayx}},{\frac {ax-cz}{zax}} \right ) +yk \left ( \left ( ax-cz \right ) by\ln \left ( {\frac {ax}{y}} \right ) -cz\ln \left ( {\frac {ax}{z}} \right ) \left ( ax-by \right ) \right ) xz \right ) }\]

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