6.9.3 2.3

6.9.3.1 [1936] Problem 1

problem number 1936

Added Jan 16, 2020.

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

Solve for $w(x,y,z)$ $$w_x+(a_1\sqrt{x}+a_0)w_y+(b_1\sqrt{x}+b_0)w_z=cw+s_1\sqrt{x}+s_0$$

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]];
 

$$\left\{\{w(x,y,z)\to e^{cx}{c}_1(-\text{a0}x-\frac{2}{3}\text{a1}{x}^{3/2}+y,-\text{b0}x-\frac{2}{3}\text{b1}{x}^{3/2}+z)+\frac{\sqrt{\pi }\text{s1}{e}^{cx}\text{Erf}\left(\sqrt{c}\sqrt{x}\right)}{2c^{3/2}}-\frac{\text{s0}+\text{s1}\sqrt{x}}{c}\}\right\}$$

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'));
 

$$w(x,y,z)=1/2\frac{1}{c}((2\mathit{\_}\mathit{F}\mathit{1}(-2/3x^{3/2}{a}_1-a_{0}x+y,-2/3x^{3/2}{b}_1-b_{0}x+z)c\sqrt{\frac{c}{\pi }}+s_1\mathrm{erf}\left(\sqrt{c}\sqrt{x}\right)){\mathrm{e}}^{cx}-2\sqrt{\frac{c}{\pi }}(s_1\sqrt{x}+s_0))\frac{1}{\sqrt{\frac{c}{\pi }}}$$

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6.9.3.2 [1937] Problem 2

problem number 1937

Added Jan 16, 2020.

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

Solve for $w(x,y,z)$ $$w_x+(b_1{x}^2+b_0)w_y+(c_1{y}^3+c_0)w_z=aw+s_1{x}^3+s_0$$

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]];
 

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

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'));
 

$$w(x,y,z)=\frac{1}{a^4}({\mathrm{e}}^{ax}\mathit{\_}\mathit{F}\mathit{1}(-1/3b_1{x}^3-b_{0}x+y,\frac{3c_{1}x}{140}({b_1}^3{x}^9+13/2{b_1}^2{x}^7{b}_0-10{b_1}^2{x}^{6}y+\frac{133b_1{x}^5{b_0}^2}{9}-42b_1{x}^4{b}_{0}y+(35y^2{b}_1+\frac{35{b_0}^3}{3})x^3-\frac{140{b_0}^2{x}^{2}y}{3}+70b_{0}x{y}^2-\frac{140y^3}{3})-c_{0}x+z)a^4+(-s_1{x}^3-s_0)a^3-3s_1{x}^2{a}^2-6s_{1}xa-6s_1)$$

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6.9.3.3 [1938] Problem 3

problem number 1938

Added Jan 16, 2020.

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

Solve for $w(x,y,z)$ $$w_x+(ay+k{x}^3)w_y+(bz+n{x}^3)w_z=cw+s{x}^2$$

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]];
 

$$\left\{\{w(x,y,z)\to e^{cx}{c}_1(\frac{e^{-ax}(a^{4}y+k(a^3{x}^3+3a^2{x}^2+6ax+6))}{a^4},\frac{e^{-bx}(b^{4}z+n(b^3{x}^3+3b^2{x}^2+6bx+6))}{b^4})-\frac{s(c^2{x}^2+2cx+2)}{c^3}\}\right\}$$

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'));
 

$$w(x,y,z)=\frac{1}{c^3}({\mathrm{e}}^{cx}\mathit{\_}\mathit{F}\mathit{1}(\frac{(k{x}^3{a}^3+a^{4}y+3k{x}^2{a}^2+6kxa+6k){\mathrm{e}}^{-ax}}{a^4},\frac{(n{x}^3{b}^3+b^{4}z+3x^{2}n{b}^2+6nxb+6n){\mathrm{e}}^{-bx}}{b^4})c^3-s(x^2{c}^2+2cx+2))$$

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6.9.3.4 [1939] Problem 4

problem number 1939

Added Jan 16, 2020.

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

Solve for $w(x,y,z)$ $$w_x+(a_{1}xy+a_2{x}^3)w_y+(b_{1}yz+b_2{y}^3)w_z=(c_{1}z+c_{2}y)w+s_1{x}^{2}y+s_{2}x{z}^2$$

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

Added Jan 16, 2020.

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

Solve for $w(x,y,z)$ $$a{x}^3{w}_x+b{y}^3{w}_y+c{z}^3{w}_z=xw+kx+s$$

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]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{x^2}{2a}}{c}_1(-\frac{bx}{a}-\frac{1}{2y^2},-\frac{cx}{a}-\frac{1}{2z^2})+\frac{\sqrt{\frac{\pi }{2}}s{e}^{\frac{x^2}{2a}}\text{Erf}\left(\frac{x}{\sqrt{2}\sqrt{a}}\right)}{\sqrt{a}}-k\}\right\}$$

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'));
 

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

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