6.9.3 2.3

6.9.3.1 [1935] Problem 1

problem number 1935

Added Jan 16, 2020.

Problem Chapter 9.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a_1 \sqrt x+ a_0) w_y + (b_1 \sqrt x+b_0) w_z = c w + s_1 \sqrt x+s_0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*Sqrt[x]+a0)*D[w[x,y,z],y]+(b1*Sqrt[x]+b0)*D[w[x,y,z],z]==c*w[x,y,z]+ s1*Sqrt[x]+s0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*sqrt(x)+a__0)*diff(w(x,y,z),y)+ (b__1*sqrt(x)+b__0)*diff(w(x,y,z),z)=c*w(x,y,z)+ s__1*sqrt(x)+s__0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.9.3.2 [1936] Problem 2

problem number 1936

Added Jan 16, 2020.

Problem Chapter 9.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (b_1 x^2+ b_0) w_y + (c_1 y^3+c_0) w_z = a w + s_1 x^3+s_0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(b1*x^2+b0)*D[w[x,y,z],y]+(c1*y^3+c0)*D[w[x,y,z],z]==a*w[x,y,z]+ s1*x^3+s0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to -\frac {a^4 \left (-e^{a x}\right ) c_1\left (-\text {b0} x-\frac {\text {b1} x^3}{3}+y,\frac {1}{4} \text {b0}^3 \text {c1} x^4+\text {b0}^2 \left (\frac {19}{60} \text {b1} \text {c1} x^6-\text {c1} x^3 y\right )+\frac {3}{280} \text {b0} \text {c1} x^2 \left (13 \text {b1}^2 x^6-84 \text {b1} x^3 y+140 y^2\right )+\frac {3}{140} \text {b1}^3 \text {c1} x^{10}-\frac {3}{14} \text {b1}^2 \text {c1} x^7 y+\frac {3}{4} \text {b1} \text {c1} x^4 y^2-\text {c0} x-\text {c1} x y^3+z\right )+a^3 \left (\text {s0}+\text {s1} x^3\right )+3 a^2 \text {s1} x^2+6 a \text {s1} x+6 \text {s1}}{a^4}\right \}\right \}\]

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^3+c__0)*diff(w(x,y,z),z)=a*w(x,y,z)+ s__1*x^3+s__0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {{\mathrm e}^{a x} f_{1} \left (-\frac {1}{3} b_{1} x^{3}-b_{0} x +y , \frac {3 \left (b_{1}^{3} x^{9}+\frac {13 b_{1}^{2} x^{7} b_{0}}{2}-10 b_{1}^{2} x^{6} y +\frac {133 b_{1} x^{5} b_{0}^{2}}{9}-42 b_{1} x^{4} b_{0} y +35 \left (y^{2} b_{1} +\frac {b_{0}^{3}}{3}\right ) x^{3}-\frac {140 b_{0}^{2} x^{2} y}{3}+70 b_{0} x \,y^{2}-\frac {140 y^{3}}{3}\right ) x c_{1}}{140}-c_{0} x +z \right ) a^{4}+\left (-s_{1} x^{3}-s_{0} \right ) a^{3}-3 s_{1} x^{2} a^{2}-6 s_{1} x a -6 s_{1}}{a^{4}}\]

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6.9.3.3 [1937] Problem 3

problem number 1937

Added Jan 16, 2020.

Problem Chapter 9.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+ k x^3) w_y + (b z+n x^3) w_z = c w + s x^2 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a*y+k*x^3)*D[w[x,y,z],y]+(b*z+n*x^3)*D[w[x,y,z],z]==c*w[x,y,z]+ s*x^2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to -\frac {s \left (c^2 x^2+2 c x+2\right )}{c^3}+e^{c x} c_1\left (\frac {e^{-a x} \left (a^4 y+k \left (a^3 x^3+3 a^2 x^2+6 a x+6\right )\right )}{a^4},\frac {e^{-b x} \left (b^4 z+n \left (b^3 x^3+3 b^2 x^2+6 b x+6\right )\right )}{b^4}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a*y+k*x^3)*diff(w(x,y,z),y)+ (b*z+n*x^3)*diff(w(x,y,z),z)=c*w(x,y,z)+ s*x^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.9.3.4 [1938] Problem 4

problem number 1938

Added Jan 16, 2020.

Problem Chapter 9.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a_1 x y+ a_2 x^3) w_y + (b_1 y z+b_2 y^3) w_z = (c_1 z+c_2 y) w + s_1 x^2 y+ s_2 x z^2 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x*y+a2*x^3)*D[w[x,y,z],y]+(b1*y*z+b2*y^3)*D[w[x,y,z],z]==(c1*z+c2*y)*w[x,y,z]+ s1*x^2*y+s2*x*z^2; 
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__1*x*y+a__2*x^3)*diff(w(x,y,z),y)+ (b__1*y*z+b__2*y^3)*diff(w(x,y,z),z)=(c__1*z+c__2*y)*w(x,y,z)+ s__1*x^2*y+s__2*x*z^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.9.3.5 [1939] Problem 5

problem number 1939

Added Jan 16, 2020.

Problem Chapter 9.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^3 w_x + b y^3 w_y + c z^3 w_z = x w + k x+ s \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*y^3*D[w[x,y,z],y]+c*z^3*D[w[x,y,z],z]==x*w[x,y,z]+ k*x+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde :=  a*diff(w(x,y,z),x)+ b*y^3*diff(w(x,y,z),y)+ c*z^3*diff(w(x,y,z),z)=x*w(x,y,z)+ k*x+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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