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]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^xK[1] (y+a (K[1]-x)) (z+b (K[1]-x))dK[1]+c_1(y-a x,z-b x)\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \frac {a b \,x^{4}}{12}-\frac {\left (a z +b y \right ) x^{3}}{6}+\frac {x^{2} y z}{2}+f_{1} \left (-a x +y , -b x +z \right )\]

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y , z\right ) = \int _{}^{y}\frac {\sqrt {\frac {x \sqrt {a b}+\ln \left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}}\, c +s}{\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}d \textit {\_a} +f_{1} \left (\frac {a \,z^{2}-b \,y^{2}}{a}, \frac {x \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\]

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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]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {a k x^2 y z K[1]}{(b y (x-K[1])+a x K[1]) (c z (x-K[1])+a x K[1])}dK[1]+c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \}\]

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'));
 

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

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