6.9.2 2.2

6.9.2.1 [1928] Problem 1

problem number 1928

Added Jan 6, 2020.

Problem Chapter 9.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a_1 x^2+a_0) w_y + (b_1 x^2+b_0)w_z = (c_1 x+c_0) w + s_1 x^2+s_0 \]

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]==(c1*x+c0)*w[x,y,z]+s1*x^2+s0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (-\text {a0} x-\frac {\text {a1} x^3}{3}+y,-\text {b0} x-\frac {\text {b1} x^3}{3}+z\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) \left (\text {c0}^2 \text {s1}+\text {c1}^2 \text {s0}+\text {c1} \text {s1}\right )}{\text {c1}^{5/2}}+\frac {\text {s1} (\text {c0}-\text {c1} x)}{\text {c1}^2}\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*x^2+a__0)*diff(w(x,y,z),y)+ (b__1*x^2+b__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x^2+s_0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = -\frac {-f_{1} \left (-\frac {1}{3} a_{1} x^{3}-a_{0} x +y , -\frac {1}{3} b_{1} x^{3}-b_{0} x +z \right ) c_{1}^{{5}/{2}} {\mathrm e}^{\frac {x \left (c_{1} x +2 c_{0} \right )}{2}}-\frac {\left (\left (c_{0}^{2}+c_{1} \right ) s_{1} +\textit {s\_0} \,c_{1}^{2}\right ) \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (c_{1} x +c_{0} \right )}{2 \sqrt {c_{1}}}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (c_{1} x +c_{0} \right )^{2}}{2 c_{1}}}}{2}+\left (x \,c_{1}^{{3}/{2}}-\sqrt {c_{1}}\, c_{0} \right ) s_{1}}{c_{1}^{{5}/{2}}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.2.2 [1929] Problem 2

problem number 1929

Added Jan 6, 2020.

Problem Chapter 9.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (b_1 x^2+b_0) w_y + (c_1 y^2+c_0)w_z = a w + s_1 x^2+s_0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(b1*x^2+b0)*D[w[x,y,z],y]+(c1*y^2+c0)*D[w[x,y,z],z]==a*w[x,y,z]+s1*x^2+s0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \frac {a^3 e^{a x} c_1\left (-\text {b0} x-\frac {\text {b1} x^3}{3}+y,-\frac {1}{3} \text {b0}^2 \text {c1} x^3-\frac {3}{10} \text {b0} \text {b1} \text {c1} x^5+\text {b0} \text {c1} x^2 y-\frac {1}{14} \text {b1}^2 \text {c1} x^7+\frac {1}{2} \text {b1} \text {c1} x^4 y-\text {c0} x-\text {c1} x y^2+z\right )-a^2 \left (\text {s0}+\text {s1} x^2\right )-2 a \text {s1} x-2 \text {s1}}{a^3}\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (b__1*x^2+b__0)*diff(w(x,y,z),y)+ (c__1*y^2+c__0)*diff(w(x,y,z),z)=a*w(x,y,z)+s__1*x^2+s_0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {{\mathrm e}^{a x} f_{1} \left (-\frac {1}{3} b_{1} x^{3}-b_{0} x +y , -\frac {\left (b_{1}^{2} x^{6}+\frac {21}{5} b_{1} x^{4} b_{0} -7 b_{1} x^{3} y +\frac {14}{3} b_{0}^{2} x^{2}-14 b_{0} x y +14 y^{2}\right ) x c_{1}}{14}-c_{0} x +z \right ) a^{3}+\left (-s_{1} x^{2}-\textit {s\_0} \right ) a^{2}-2 s_{1} x a -2 s_{1}}{a^{3}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.2.3 [1930] Problem 3

problem number 1930

Added Jan 6, 2020.

Problem Chapter 9.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+k_1 x^2+k_0) w_y + (b z+n_1 x^2+n_0)w_z = (c_1 x+c_0) w + s_1 x+s_0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a*y+k1*x^2+k0)*D[w[x,y,z],y]+(b*z+n1*x^2+n0)*D[w[x,y,z],z]==(c1*x+c0)*w[x,y,z]+s1*x+s0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (\frac {e^{-a x} \left (a^3 y+a^2 \left (\text {k0}+\text {k1} x^2\right )+2 a \text {k1} x+2 \text {k1}\right )}{a^3},\frac {e^{-b x} \left (b^3 z+b^2 \left (\text {n0}+\text {n1} x^2\right )+2 b \text {n1} x+2 \text {n1}\right )}{b^3}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a*y+k__1*x^2+k__0)*diff(w(x,y,z),y)+ (b*z+n__1*x^2+n__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x+s_0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = -\frac {-2 f_{1} \left (\frac {\left (y \,a^{3}+\left (k_{1} x^{2}+k_{0} \right ) a^{2}+2 k_{1} x a +2 k_{1} \right ) {\mathrm e}^{-a x}}{a^{3}}, \frac {\left (z \,b^{3}+\left (n_{1} x^{2}+n_{0} \right ) b^{2}+2 n_{1} x b +2 n_{1} \right ) {\mathrm e}^{-b x}}{b^{3}}\right ) c_{1}^{{3}/{2}} {\mathrm e}^{\frac {x \left (c_{1} x +2 c_{0} \right )}{2}}+{\mathrm e}^{\frac {\left (c_{1} x +c_{0} \right )^{2}}{2 c_{1}}} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (c_{1} x +c_{0} \right )}{2 \sqrt {c_{1}}}\right ) \left (c_{0} s_{1} -c_{1} \textit {s\_0} \right )+2 s_{1} \sqrt {c_{1}}}{2 c_{1}^{{3}/{2}}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.2.4 [1931] Problem 4

problem number 1931

Added Jan 6, 2020.

Problem Chapter 9.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a_2 x y+a_1 x+a_0) w_y + (b_3 y z+b_2 y^2+b_1 x^2+b_0)w_z = (c_3 z+c_2 y+c_1 x+c_0) w + s_1 x y+s_2 x z \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a2*x*y+a1*x+a0)*D[w[x,y,z],y]+(b3*y*z+b2*y^2+b1*x^2+b0)*D[w[x,y,z],z]==(c3*z+c2*y+c1*x+c0)*w[x,y,z]+s1*x*y+s2*x*z; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$Aborted

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__2*x*y+a__1*x+a__0)*diff(w(x,y,z),y)+ (b__3*y*z+b__2*y^2+b__1*x^2+b__0)*diff(w(x,y,z),z)=(c__3*z+c__2*y+c__1*x+c__0)*w(x,y,z)+s__1*x*y+s__2*x*z; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[\text {Expression too large to display}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.2.5 [1932] Problem 5

problem number 1932

Added Jan 6, 2020.

Problem Chapter 9.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.2.6 [1933] Problem 6

problem number 1933

Added Jan 6, 2020.

Problem Chapter 9.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.2.7 [1934] Problem 7

problem number 1934

Added Jan 6, 2020.

Problem Chapter 9.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^2 w_x + b y^2 w_y + c z^2 w_z = (k x+s) w + p x+q \]

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

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

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________