6.9.2 2.2

6.9.2.1 [1929] Problem 1

problem number 1929

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]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{1}{2}x(2\text{c0}+\text{c1}x)}{c}_1(-\text{a0}x-\frac{\text{a1}{x}^3}{3}+y,-\text{b0}x-\frac{\text{b1}{x}^3}{3}+z)+\frac{\sqrt{\frac{\pi }{2}}{e}^{\frac{(\text{c0}+\text{c1}x{)}^2}{2\text{c1}}}\text{Erf}\left(\frac{\text{c0}+\text{c1}x}{\sqrt{2}\sqrt{\text{c1}}}\right)({\text{c0}}^2\text{s1}+{\text{c1}}^2\text{s0}+\text{c1}\text{s1})}{{\text{c1}}^{5/2}}+\frac{\text{s1}(\text{c0}-\text{c1}x)}{{\text{c1}}^2}\}\right\}$$

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'));
 

$$w(x,y,z)=\frac{{\mathrm{e}}^{1/2x(c_{1}x+2c_0)}}{{c_1}^{9/2}}({c_1}^{9/2}\mathit{\_}\mathit{F}\mathit{1}(-1/3a_1{x}^3-a_{0}x+y,-1/3b_1{x}^3-b_{0}x+z)+1/2\sqrt{\pi }{\mathrm{e}}^{1/2\frac{{c_0}^2}{c_1}}\sqrt{2}{c_1}^2({c_0}^2{s}_1+{c_1}^2\mathit{s}\mathit{\_}\mathit{0}+c_1{s}_1)\mathrm{erf}\left(1/2\sqrt{2}(x\sqrt{c_1}+\frac{c_0}{\sqrt{c_1}})\right)+({c_1}^{5/2}{c}_0-x{c_1}^{7/2}){\mathrm{e}}^{-1/2c_1{x}^2-c_{0}x}{s}_1)$$

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6.9.2.2 [1930] Problem 2

problem number 1930

Added Jan 6, 2020.

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

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

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]];
 

$$\left\{\{w(x,y,z)\to \frac{a^3{e}^{ax}{c}_1(-\text{b0}x-\frac{\text{b1}{x}^3}{3}+y,-\frac{1}{3}{\text{b0}}^2\text{c1}{x}^3-\frac{3}{10}\text{b0}\text{b1}\text{c1}{x}^5+\text{b0}\text{c1}{x}^{2}y-\frac{1}{14}{\text{b1}}^2\text{c1}{x}^7+\frac{1}{2}\text{b1}\text{c1}{x}^{4}y-\text{c0}x-\text{c1}x{y}^2+z)-a^2(\text{s0}+\text{s1}{x}^2)-2a\text{s1}x-2\text{s1}}{a^3}\}\right\}$$

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'));
 

$$w(x,y,z)=\frac{1}{a^3}({\mathrm{e}}^{ax}\mathit{\_}\mathit{F}\mathit{1}(-1/3b_1{x}^3-b_{0}x+y,-1/14({b_1}^2{x}^6+\frac{21b_1{x}^4{b}_0}{5}-7b_1{x}^{3}y+14/3{b_0}^2{x}^2-14x{b}_{0}y+14y^2)x{c}_1-c_{0}x+z)a^3+(-s_1{x}^2-\mathit{s}\mathit{\_}\mathit{0})a^2-2s_{1}xa-2s_1)$$

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6.9.2.3 [1931] Problem 3

problem number 1931

Added Jan 6, 2020.

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

Solve for $w(x,y,z)$ $$w_x+(ay+k_1{x}^2+k_0)w_y+(bz+n_1{x}^2+n_0)w_z=(c_{1}x+c_0)w+s_{1}x+s_0$$

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]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{1}{2}x(2\text{c0}+\text{c1}x)}{c}_1(\frac{e^{-ax}(a^{3}y+a^2(\text{k0}+\text{k1}{x}^2)+2a\text{k1}x+2\text{k1})}{a^3},\frac{e^{-bx}(b^{3}z+b^2(\text{n0}+\text{n1}{x}^2)+2b\text{n1}x+2\text{n1})}{b^3})+\frac{\sqrt{\frac{\pi }{2}}{e}^{\frac{(\text{c0}+\text{c1}x{)}^2}{2\text{c1}}}\text{Erf}\left(\frac{\text{c0}+\text{c1}x}{\sqrt{2}\sqrt{\text{c1}}}\right)(\text{c1}\text{s0}-\text{c0}\text{s1})}{{\text{c1}}^{3/2}}-\frac{\text{s1}}{\text{c1}}\}\right\}$$

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'));
 

$$w(x,y,z)=-1/2\frac{{\mathrm{e}}^{1/2x(c_{1}x+2c_0)}}{{c_1}^{5/2}}(-2\mathit{\_}\mathit{F}\mathit{1}(\frac{(y{a}^3+(k_1{x}^2+k_0)a^2+2k_{1}xa+2k_1){\mathrm{e}}^{-ax}}{a^3},\frac{(z{b}^3+(n_1{x}^2+n_0)b^2+2n_{1}xb+2n_1){\mathrm{e}}^{-bx}}{b^3}){c_1}^{5/2}+\sqrt{2}\sqrt{\pi }{\mathrm{e}}^{1/2\frac{{c_0}^2}{c_1}}{c}_1(c_0{s}_1-c_1\mathit{s}\mathit{\_}\mathit{0})\mathrm{erf}\left(1/2\sqrt{2}(x\sqrt{c_1}+\frac{c_0}{\sqrt{c_1}})\right)+2{\mathrm{e}}^{-1/2c_1{x}^2-c_{0}x}{c_1}^{3/2}{s}_1)$$

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6.9.2.4 [1932] Problem 4

problem number 1932

Added Jan 6, 2020.

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

Solve for $w(x,y,z)$ $$w_x+(a_{2}xy+a_{1}x+a_0)w_y+(b_{3}yz+b_2{y}^2+b_1{x}^2+b_0)w_z=(c_{3}z+c_{2}y+c_{1}x+c_0)w+s_{1}xy+s_{2}xz$$

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'));
 

$$\text{Expression too large to display}$$

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6.9.2.5 [1933] Problem 5

problem number 1933

Added Jan 6, 2020.

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

Solve for $w(x,y,z)$ $$ax{w}_x+bx{w}_y+cz{w}_z=kxw+s{x}^2$$

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]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{kx}{a}}{c}_1(y-\frac{bx}{a},z{x}^{-\frac{c}{a}})-\frac{s(a+kx)}{k^2}\}\right\}$$

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'));
 

$$w(x,y,z)=\frac{1}{k^2}({\mathrm{e}}^{\frac{kx}{a}}\mathit{\_}\mathit{F}\mathit{1}(\frac{ay-bx}{a},z{x}^{-\frac{c}{a}})k^2-s(kx+a))$$

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6.9.2.6 [1934] Problem 6

problem number 1934

Added Jan 6, 2020.

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

Solve for $w(x,y,z)$ $$ax{w}_x+by{w}_y+cz{w}_z=kxw+s{x}^2$$

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]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{kx}{a}}{c}_1(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}})-\frac{s(a+kx)}{k^2}\}\right\}$$

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'));
 

$$w(x,y,z)=\frac{1}{k^2}({\mathrm{e}}^{\frac{kx}{a}}\mathit{\_}\mathit{F}\mathit{1}(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}})k^2-s(kx+a))$$

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6.9.2.7 [1935] Problem 7

problem number 1935

Added Jan 6, 2020.

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

Solve for $w(x,y,z)$ $$a{x}^2{w}_x+b{y}^2{w}_y+c{z}^2{w}_z=(kx+s)w+px+q$$

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]];
 

$$\left\{\{w(x,y,z)\to \frac{e^{-\frac{s}{ax}}{(-\frac{s}{ax})}^{-\frac{k}{a}}(as{x}^{\frac{k}{a}}{(-\frac{s}{ax})}^{\frac{k}{a}}{c}_1(\frac{b}{ax}-\frac{1}{y},\frac{c}{ax}-\frac{1}{z})+ps\text{Gamma}(\frac{k}{a},-\frac{s}{ax})-aq\text{Gamma}(\frac{a+k}{a},-\frac{s}{ax}))}{as}\}\right\}$$

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'));
 

$$w(x,y,z)=(\int \frac{px+q}{a}{x}^{\frac{-2a-k}{a}}{\mathrm{e}}^{\frac{s}{ax}}\mathrm{d}x+\mathit{\_}\mathit{F}\mathit{1}(\frac{ax-by}{xya},\frac{ax-cz}{xza})){\mathrm{e}}^{-\frac{s}{ax}}{x}^{\frac{k}{a}}$$

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