6.8.2 2.2

6.8.2.1 [1759] Problem 1

problem number 1759

Added June 28, 2019.

Problem Chapter 8.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c w_z = (\lambda x^2 +\beta y^2+\gamma z^2+\delta ) w \]

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

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

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

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

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6.8.2.2 [1760] Problem 2

problem number 1760

Added June 28, 2019.

Problem Chapter 8.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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6.8.2.3 [1761] Problem 3

problem number 1761

Added June 28, 2019.

Problem Chapter 8.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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6.8.2.4 [1762] Problem 4

problem number 1762

Added June 28, 2019.

Problem Chapter 8.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\text {a0} \text {b2} x}{\text {a1}}-\frac {\text {b2} y}{\text {a1}}-\text {b0} x-\frac {\text {b1} x^3}{3}+\frac {\text {b2} x^3}{3}+z,e^{-\frac {\text {a1} x^2}{2}} (x+y)-\frac {\sqrt {\frac {\pi }{2}} (\text {a0}+1) \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\sqrt {\text {a1}}}\right ) \exp \left (\frac {2 (\text {a0}+1) \text {a1} \text {b2} \text {c1} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {\text {a1} x^2}{2}\right )+\text {a1} x \left (2 \text {b2} \text {c1} ((\text {a0}-1) x-2 y)+\text {a1} \left (-2 \text {b0} \text {c1} x-\text {b1} \text {c1} x^3+\text {b2} \text {c1} x^3+2 \text {c0} x+4 \text {c1} z+2 \text {c2} x+4 s\right )\right )+2 \text {b2} \text {c1} e^{-\frac {\text {a1} x^2}{2}} \text {erfi}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right ) \left (\sqrt {2 \pi } \sqrt {\text {a1}} (x+y)-\pi (\text {a0}+1) e^{\frac {\text {a1} x^2}{2}} \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )\right )}{4 \text {a1}^2}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {{\mathrm e}^{-\frac {\operatorname {a1} \,x^{2}}{2}} \sqrt {\operatorname {a1}}\, \left (y +x \right )-\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\operatorname {a1}}\, x}{2}\right ) \left (\operatorname {a0} +1\right )}{2}}{\sqrt {\operatorname {a1}}}, \frac {\left (\left (-\operatorname {b1} +\operatorname {b2} \right ) x^{3}-3 \operatorname {b0} x +3 z \right ) \operatorname {a1} +3 \operatorname {b2} \left (x \operatorname {a0} -y \right )}{3 \operatorname {a1}}\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (6 \operatorname {b2} \operatorname {c1} \sqrt {\operatorname {a1}}\, \left (y +x \right ) {\mathrm e}^{\frac {\operatorname {a1} \left (\textit {\_a}^{2}-x^{2}\right )}{2}}-3 \operatorname {b2} \operatorname {c1} \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {a0} +1\right ) \left (\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\operatorname {a1}}\, x}{2}\right )-\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\operatorname {a1}}\, \textit {\_a}}{2}\right )\right ) {\mathrm e}^{\frac {\operatorname {a1} \,\textit {\_a}^{2}}{2}}+\left (\left (\left (-2 \textit {\_a}^{3}+2 x^{3}\right ) \operatorname {b2} -2 \operatorname {b1} \,x^{3}+2 \textit {\_a}^{3} \operatorname {b1} -6 \operatorname {b0} x +6 \textit {\_a} \operatorname {b0} +6 z \right ) \operatorname {c1} +\left (6 \operatorname {c0} +6 \operatorname {c2} \right ) \textit {\_a} +6 s \right ) \operatorname {a1}^{{3}/{2}}+6 \sqrt {\operatorname {a1}}\, \operatorname {b2} \left (\left (-\operatorname {a0} -1\right ) \textit {\_a} +x \operatorname {a0} -y \right ) \operatorname {c1} \right )d \textit {\_a}}{6 \operatorname {a1}^{{3}/{2}}}}\]

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6.8.2.5 [1763] Problem 5

problem number 1763

Added June 28, 2019.

Problem Chapter 8.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*(lambda*x+beta*y+gama*z)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

\[w \left (x , y , z\right ) = f_{1} \left (y \,x^{-\frac {b}{a}}, z \,x^{-\frac {c}{a}}\right ) {\mathrm e}^{\frac {x \left (\left (2 \beta y +2 \operatorname {gama} z +\lambda x \right ) a^{2}+\left (\lambda \left (b +c \right ) x +2 b z \operatorname {gama} +2 c y \beta \right ) a +b c \lambda x \right )}{2 a \left (a +b \right ) \left (a +c \right )}}\]

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6.8.2.6 [1764] Problem 6

problem number 1764

Added June 28, 2019.

Problem Chapter 8.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^2 w_x + b x y w_y + c x z w_z = (\lambda x+\beta y+\gamma z) w \]

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]== (lambda*x+beta*y+gama*z)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to x^{\frac {\lambda }{a}} e^{-\frac {\frac {\beta y}{a-b}+\frac {\text {gama} z}{a-c}}{x}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (y \,x^{-\frac {b}{a}}, z \,x^{-\frac {c}{a}}\right ) x^{\frac {\lambda }{a}} {\mathrm e}^{-\frac {\frac {\beta y}{a -b}+\frac {\operatorname {gama} z}{a -c}}{x}}\]

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6.8.2.7 [1765] Problem 7

problem number 1765

Added June 28, 2019.

Problem Chapter 8.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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6.8.2.8 [1766] Problem 8

problem number 1766

Added June 28, 2019.

Problem Chapter 8.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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6.8.2.9 [1767] Problem 9

problem number 1767

Added June 28, 2019.

Problem Chapter 8.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^2 w_x + b y^2 w_y + c z^2 w_z = (\lambda x^2+\beta y^2 + \gamma z^2) w \]

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

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

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

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

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