6.7.2 2.2

6.7.2.1 [1589] Problem 1

problem number 1589

Added June 1, 2019.

Problem Chapter 7.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c w_z = \alpha x^2+\beta y^2+\gamma z^2 + \delta \]

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]==alpha*x^2+beta*y^2+gamma*z^2+delta; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \frac {1}{3} \left (3 c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )+\frac {\alpha x^3+3 \delta x}{a}+\frac {\beta y^3}{b}+\frac {\gamma z^3}{c}\right )\right \}\right \}\]

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

\[w \left ( x,y,z \right ) =1/24\,{\frac {1}{{a}^{3}} \left ( 24\,{\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {za-cx}{a}} \right ) {a}^{3}+8\,x \left ( \left ( \alpha \,{x}^{2}+3\,\beta \,{y}^{2}+3/8\,{z}^{2}+3\,\delta \right ) {a}^{2}-3\,x \left ( b\beta \,y+1/8\,cz \right ) a+{x}^{2} \left ( {b}^{2}\beta +1/8\,{c}^{2} \right ) \right ) \right ) }\]

____________________________________________________________________________________

6.7.2.2 [1590] Problem 2

problem number 1590

Added June 1, 2019.

Problem Chapter 7.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a_1 x^2+a_0) w_y + (b_1 x^2+b_0) w_z = \alpha x+\beta y+\gamma z + \delta \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x^2+a0)*D[w[x, y,z], y] +(b1*x^2+b0)*D[w[x,y,z],z]==alpha*x+beta*y+gamma*z+delta; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\text {a0} x-\frac {\text {a1} x^3}{3}+y,-\text {b0} x-\frac {\text {b1} x^3}{3}+z\right )-\frac {1}{4} x \left (2 \text {a0} \beta x+\text {a1} \beta x^3-2 \alpha x+2 \text {b0} \gamma x+\text {b1} \gamma x^3-4 \beta y-4 \delta -4 \gamma z\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+  (a1*x^2+a0)*diff(w(x,y,z),y)+(b1*x^2+b0)*diff(w(x,y,z),z)=alpha*x+beta*y+gamma*z+delta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -1/3\,{\it a1}\,{x}^{3}-{\it a0}\,x+y,-1/3\,{\it b1}\,{x}^{3}-{\it b0}\,x+z \right ) +1/32\, \left ( -8\,\beta \,{\it a1}-{\it b1} \right ) {x}^{4}+1/32\, \left ( -16\,\beta \,{\it a0}+16\,\alpha -2\,{\it b0} \right ) {x}^{2}+1/32\, \left ( 32\,\beta \,y+32\,\delta +4\,z \right ) x\]

____________________________________________________________________________________

6.7.2.3 [1591] Problem 3

problem number 1591

Added June 1, 2019.

Problem Chapter 7.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+k_1 x^2+k_0) w_y + (b z+s_1 x^2+s_0) w_z = c_1 x^2+c_0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*y+k1*x^2+k0)*D[w[x, y,z], y] +(b*z+s1*x^2+s0)*D[w[x,y,z],z]==c1*x^2+c0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-a x} \left (a^3 y+a^2 \left (\text {k0}+\text {k1} x^2\right )+2 a \text {k1} x+2 \text {k1}\right )}{a^3},\frac {e^{-b x} \left (b^3 z+b^2 \left (\text {s0}+\text {s1} x^2\right )+2 b \text {s1} x+2 \text {s1}\right )}{b^3}\right )+\text {c0} x+\frac {\text {c1} x^3}{3}\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+  (a*y+k1*x^2+k0)*diff(w(x,y,z),y)+(b*z+s1*x^2+s0)*diff(w(x,y,z),z)=c1*x^2+c0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =1/3\,{\it c1}\,{x}^{3}+{\it c0}\,x+{\it \_F1} \left ( {\frac { \left ( {a}^{3}y+ \left ( {\it k1}\,{x}^{2}+{\it k0} \right ) {a}^{2}+2\,{\it k1}\,xa+2\,{\it k1} \right ) {{\rm e}^{-ax}}}{{a}^{3}}},{\frac { \left ( z{b}^{3}+ \left ( {\it s1}\,{x}^{2}+{\it s0} \right ) {b}^{2}+2\,{\it s1}\,xb+2\,{\it s1} \right ) {{\rm e}^{-bx}}}{{b}^{3}}} \right ) \]

____________________________________________________________________________________

6.7.2.4 [1592] Problem 4

problem number 1592

Added June 1, 2019.

Problem Chapter 7.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a_2 x y+a_1 x^2+a_0) w_y + (b_2 x y+b_1 x^2+b_0) w_z = c_2 y+c_1 z+c_0 x+s \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a2*x+a1*x^2+a0)*D[w[x, y,z], y] +(b2*x*y+b1*x^2+b0)*D[w[x,y,z],z]==c2*y+c1*z+c0*x+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\text {a0} x-\frac {\text {a1} x^3}{3}-\frac {\text {a2} x^2}{2}+y,\frac {1}{6} \text {a0} \text {b2} x^3+\frac {1}{10} \text {a1} \text {b2} x^5+\frac {1}{8} \text {a2} \text {b2} x^4-\text {b0} x-\frac {\text {b1} x^3}{3}-\frac {1}{2} \text {b2} x^2 y+z\right )+\frac {1}{180} x \left (15 \text {a0} \text {b2} \text {c1} x^3-90 \text {a0} \text {c2} x+10 \text {a1} \text {b2} \text {c1} x^5-45 \text {a1} \text {c2} x^3+12 \text {a2} \text {b2} \text {c1} x^4-60 \text {a2} \text {c2} x^2-90 \text {b0} \text {c1} x-45 \text {b1} \text {c1} x^3-60 \text {b2} \text {c1} x^2 y+90 \text {c0} x+180 \text {c1} z+180 \text {c2} y+180 s\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+ (a2*x+a1*x^2+a0)*diff(w(x,y,z),y)+(b2*x*y+b1*x^2+b0)*diff(w(x,y,z),z)=c2*y+c1*z+c0*x+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -1/3\,{\it a1}\,{x}^{3}-1/2\,{\it a2}\,{x}^{2}-{\it a0}\,x+y,1/10\,{x}^{5}{\it b2}\,{\it a1}+1/8\,{x}^{4}{\it b2}\,{\it a2}+{\frac { \left ( 20\,{\it b2}\,{\it a0}-40\,{\it b1} \right ) {x}^{3}}{120}}-1/2\,{\it b2}\,{x}^{2}y-{\it b0}\,x+z \right ) +1/18\,{\it c1}\,{x}^{6}{\it b2}\,{\it a1}+1/15\,{\it c1}\,{x}^{5}{\it b2}\,{\it a2}+{\frac { \left ( \left ( 15\,{\it b2}\,{\it a0}-45\,{\it b1} \right ) {\it c1}-45\,{\it a1}\,{\it c2} \right ) {x}^{4}}{180}}+{\frac { \left ( -60\,{\it b2}\,{\it c1}\,y-60\,{\it c2}\,{\it a2} \right ) {x}^{3}}{180}}+{\frac { \left ( -90\,{\it c2}\,{\it a0}-90\,{\it b0}\,{\it c1}+90\,{\it c0} \right ) {x}^{2}}{180}}+{\frac { \left ( 180\,{\it c1}\,z+180\,{\it c2}\,y+180\,s \right ) x}{180}}\]

____________________________________________________________________________________

6.7.2.5 [1593] Problem 5

problem number 1593

Added June 1, 2019.

Problem Chapter 7.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y + c z w_z = x(\alpha x+\beta y+\gamma z) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )+\frac {x \left (a^2 (\alpha x+2 \beta y+2 \gamma z)+a \alpha x (b+c)+2 a (b \gamma z+\beta c y)+\alpha b c x\right )}{2 a (a+b) (a+c)}\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y,z \right ) =1/2\,{\frac {\alpha \,{x}^{2}}{a}}+ \left ( {\frac {\beta \,y}{a} \left ( {\frac {b}{a}}+1 \right ) ^{-1}}+1/8\,{\frac {z}{a} \left ( {\frac {c}{a}}+1 \right ) ^{-1}} \right ) x+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) \]

____________________________________________________________________________________

6.7.2.6 [1594] Problem 6

problem number 1594

Added June 1, 2019.

Problem Chapter 7.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )+\frac {\alpha \log (x)}{a}+\frac {-a (\beta y+\gamma z)+b \gamma z+\beta c y}{x (a-b) (a-c)}\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y,z \right ) ={\frac {\alpha \,\ln \left ( x \right ) }{a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) +{\frac {1}{x} \left ( -1/8\,{\frac {z}{a-c}}-{\frac {\beta \,y}{a-b}} \right ) }\]

____________________________________________________________________________________

6.7.2.7 [1595] Problem 7

problem number 1595

Added June 1, 2019.

Problem Chapter 7.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y,z \right ) ={\frac {1}{a-2\,b} \left ( \left ( a-2\,b \right ) {\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},{\frac {ax-cz}{axz}} \right ) -{\frac {k{y}^{2}}{x}} \right ) }\]

____________________________________________________________________________________

6.7.2.8 [1596] Problem 8

problem number 1596

Added June 1, 2019.

Problem Chapter 7.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

\[w \left ( x,y,z \right ) =-{\frac {kxy}{-ax+by}\ln \left ( {\frac {ax}{y}} \right ) }+{\it \_F1} \left ( {\frac {ax-by}{axy}},{\frac {ax-cz}{axz}} \right ) \]

____________________________________________________________________________________

6.7.2.9 [1597] Problem 9

problem number 1597

Added June 1, 2019.

Problem Chapter 7.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]==alpha*x^2+beta*y^2+gamma*z^2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

\[w \left ( x,y,z \right ) =-{\frac {\beta \,{y}^{2}}{ax-by}}+{\frac {{z}^{2}}{-8\,ax+8\,cz}}+{\frac {\alpha \,x}{a}}+{\it \_F1} \left ( {\frac {ax-by}{axy}},{\frac {ax-cz}{axz}} \right ) \]

____________________________________________________________________________________