#### 6.9.3 2.3

6.9.3.1 [1936] Problem 1
6.9.3.2 [1937] Problem 2
6.9.3.3 [1938] Problem 3
6.9.3.4 [1939] Problem 4
6.9.3.5 [1940] Problem 5

##### 6.9.3.1 [1936] Problem 1

problem number 1936

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'));



$w \left ( x,y,z \right ) =1/2\,{\frac {1}{c} \left ( \left ( 2\,{\it \_F1} \left ( -2/3\,{x}^{3/2}a_{1}-a_{0}\,x+y,-2/3\,{x}^{3/2}b_{1}-b_{0}\,x+z \right ) c\sqrt {{\frac {c}{\pi }}}+s_{1}\,\erf \left ( \sqrt {c}\sqrt {x} \right ) \right ) {{\rm e}^{cx}}-2\,\sqrt {{\frac {c}{\pi }}} \left ( s_{1}\,\sqrt {x}+s_{0} \right ) \right ) {\frac {1}{\sqrt {{\frac {c}{\pi }}}}}}$

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##### 6.9.3.2 [1937] Problem 2

problem number 1937

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'));



$w \left ( x,y,z \right ) ={\frac {1}{{a}^{4}} \left ( {{\rm e}^{ax}}{\it \_F1} \left ( -1/3\,b_{1}\,{x}^{3}-b_{0}\,x+y,{\frac {3\,c_{1}\,x}{140} \left ( {b_{1}}^{3}{x}^{9}+13/2\,{b_{1}}^{2}{x}^{7}b_{0}-10\,{b_{1}}^{2}{x}^{6}y+{\frac {133\,b_{1}\,{x}^{5}{b_{0}}^{2}}{9}}-42\,b_{1}\,{x}^{4}b_{0}\,y+ \left ( 35\,{y}^{2}b_{1}+{\frac {35\,{b_{0}}^{3}}{3}} \right ) {x}^{3}-{\frac {140\,{b_{0}}^{2}{x}^{2}y}{3}}+70\,b_{0}\,x{y}^{2}-{\frac {140\,{y}^{3}}{3}} \right ) }-c_{0}\,x+z \right ) {a}^{4}+ \left ( -s_{1}\,{x}^{3}-s_{0} \right ) {a}^{3}-3\,s_{1}\,{x}^{2}{a}^{2}-6\,s_{1}\,xa-6\,s_{1} \right ) }$

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##### 6.9.3.3 [1938] Problem 3

problem number 1938

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]];



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

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'));



$w \left ( x,y,z \right ) ={\frac {1}{{c}^{3}} \left ( {{\rm e}^{cx}}{\it \_F1} \left ( {\frac { \left ( k{x}^{3}{a}^{3}+{a}^{4}y+3\,k{x}^{2}{a}^{2}+6\,kxa+6\,k \right ) {{\rm e}^{-ax}}}{{a}^{4}}},{\frac { \left ( n{x}^{3}{b}^{3}+{b}^{4}z+3\,{x}^{2}n{b}^{2}+6\,nxb+6\,n \right ) {{\rm e}^{-bx}}}{{b}^{4}}} \right ) {c}^{3}-s \left ( {x}^{2}{c}^{2}+2\,cx+2 \right ) \right ) }$

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##### 6.9.3.4 [1939] Problem 4

problem number 1939

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 [1940] Problem 5

problem number 1940

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'));

`

$w \left ( x,y,z \right ) =1/2\, \left ( {{\rm e}^{1/2\,{\frac {{x}^{2}}{a}}}}s\erf \left ( 1/2\,{\frac {\sqrt {2}x}{\sqrt {a}}} \right ) \sqrt {2}+2\,{{\rm e}^{1/2\,{\frac {{x}^{2}}{a}}}}{\it \_F1} \left ( {\frac {2\,bx{y}^{2}+a}{a{y}^{2}}},{\frac {2\,cx{z}^{2}+a}{a{z}^{2}}} \right ) \sqrt {{\frac {a}{\pi }}}-2\,k\sqrt {{\frac {a}{\pi }}} \right ) {\frac {1}{\sqrt {{\frac {a}{\pi }}}}}$

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