6.7.2 2.2

6.7.2.1 [1588] Problem 1

problem number 1588

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'));
 

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

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6.7.2.2 [1589] Problem 2

problem number 1589

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]];
 

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

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'));
 

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

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6.7.2.3 [1590] Problem 3

problem number 1590

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} \left (\text {k1} K[1]^2+\text {k0}\right )dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} \left (\text {s1} K[2]^2+\text {s0}\right )dK[2]\right )+\text {c0} x+\frac {\text {c1} x^3}{3}\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \frac {\operatorname {c1} \,x^{3}}{3}+\operatorname {c0} x +f_{1} \left (\frac {\left (y \,a^{3}+\left (\operatorname {k1} \,x^{2}+\operatorname {k0} \right ) a^{2}+2 \operatorname {k1} x a +2 \operatorname {k1} \right ) {\mathrm e}^{-a x}}{a^{3}}, \frac {\left (z \,b^{3}+\left (\operatorname {s1} \,x^{2}+\operatorname {s0} \right ) b^{2}+2 \operatorname {s1} x b +2 \operatorname {s1} \right ) {\mathrm e}^{-b x}}{b^{3}}\right )\]

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6.7.2.4 [1591] Problem 4

problem number 1591

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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\frac {1}{3} \operatorname {a1} \,x^{3}-\frac {1}{2} \operatorname {a2} \,x^{2}-\operatorname {a0} x +y , \frac {\operatorname {b2} \,x^{5} \operatorname {a1}}{10}+\frac {\operatorname {b2} \,x^{4} \operatorname {a2}}{8}+\frac {\left (\operatorname {a0} \operatorname {b2} -2 \operatorname {b1} \right ) x^{3}}{6}-\frac {\operatorname {b2} \,x^{2} y}{2}-\operatorname {b0} x +z \right )+\frac {\operatorname {c1} \operatorname {b2} \,x^{6} \operatorname {a1}}{18}+\frac {\operatorname {c1} \operatorname {b2} \,x^{5} \operatorname {a2}}{15}+\frac {\left (\left (\operatorname {a0} \operatorname {b2} -3 \operatorname {b1} \right ) \operatorname {c1} -3 \operatorname {c2} \operatorname {a1} \right ) x^{4}}{12}-\frac {\left (\operatorname {b2} \operatorname {c1} y +\operatorname {c2} \operatorname {a2} \right ) x^{3}}{3}+\frac {\left (-\operatorname {c2} \operatorname {a0} -\operatorname {c1} \operatorname {b0} +\operatorname {c0} \right ) x^{2}}{2}+\left (\operatorname {c1} z +\operatorname {c2} y +s \right ) x\]

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6.7.2.5 [1592] Problem 5

problem number 1592

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]];
 

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

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'));
 

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

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6.7.2.6 [1593] Problem 6

problem number 1593

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'));
 

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

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6.7.2.7 [1594] Problem 7

problem number 1594

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]];
 

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

Maple ✓

restart; 
local gamma; 
pde := a*x^2*diff(w(x,y,z),x)+ b*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'));
 

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

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6.7.2.8 [1595] Problem 8

problem number 1595

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]];
 

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

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

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

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6.7.2.9 [1596] Problem 9

problem number 1596

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'));
 

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

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