7.9.1 2.1

7.9.1.1 [1921] Problem 1

problem number 1921

Added Jan 6, 2020.

Problem Chapter 9.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{w}_x+b{w}_y+c{w}_z=(\alpha x+\beta )w+px+q$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*D[w[x,y,z],z]==(alpha*x+beta)*w[x,y,z]+p*x+q; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{x(\alpha x+2\beta )}{2a}}{c}_1(y-\frac{bx}{a},z-\frac{cx}{a})+\frac{\sqrt{\frac{\pi }{2}}{e}^{\frac{(\alpha x+\beta {)}^2}{2a\alpha }}(\alpha q-\beta p)\text{erf}\left(\frac{\alpha x+\beta }{\sqrt{2}\sqrt{a}\sqrt{\alpha }}\right)}{\sqrt{a}{\alpha }^{3/2}}-\frac{p}{\alpha }\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{(-\frac{\sqrt{2}\sqrt{\pi }(-\alpha q+\beta p)\mathrm{erf}(\frac{\sqrt{2}\sqrt{\frac{\alpha }{a}}x}{2}+\frac{\sqrt{2}\beta }{2\sqrt{\frac{\alpha }{a}}a}){\mathrm{e}}^{\frac{{\beta }^2}{2a\alpha }}}{2}+\sqrt{\frac{\alpha }{a}}(\alpha \mathit{\text{\_F1}}(\frac{ay-bx}{a},\frac{az-cx}{a})-p{\mathrm{e}}^{-\frac{(\alpha x+2\beta )x}{2a}})a){\mathrm{e}}^{\frac{(\alpha x+2\beta )x}{2a}}}{\sqrt{\frac{\alpha }{a}}a\alpha }$$

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7.9.1.2 [1922] Problem 2

problem number 1922

Added Jan 6, 2020.

Problem Chapter 9.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+az{w}_y+by{w}_z=(cx+k)w+px+q$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to e^{\frac{1}{2}x(cx+2k)}{c}_1(\frac{e^{-\sqrt{a}\sqrt{b}x}(\sqrt{b}y(e^{2\sqrt{a}\sqrt{b}x}+1)-\sqrt{a}z(e^{2\sqrt{a}\sqrt{b}x}-1))}{2\sqrt{b}},\frac{e^{-\sqrt{a}\sqrt{b}x}(\sqrt{a}z(e^{2\sqrt{a}\sqrt{b}x}+1)-\sqrt{b}y(e^{2\sqrt{a}\sqrt{b}x}-1))}{2\sqrt{a}})+\frac{\sqrt{\frac{\pi }{2}}{e}^{\frac{(cx+k{)}^2}{2c}}\text{erf}\left(\frac{cx+k}{\sqrt{2}\sqrt{c}}\right)(cq-kp)}{c^{3/2}}-\frac{p}{c}\}\right\}$$

Maple ✓

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

$$w(x,y,z)=({\int }_{}^y-\frac{((-\ln \left(\frac{\mathit{\text{\_b}}ab+\sqrt{ab}\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}{\sqrt{ab}}\right)+\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right))p+(-px-q)\sqrt{ab}){\mathrm{e}}^{-\frac{\int -\frac{(-\ln \left(\frac{\mathit{\text{\_b}}ab+\sqrt{ab}\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}{\sqrt{ab}}\right)+\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right))c+(-cx-k)\sqrt{ab}}{\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}d\mathit{\text{\_b}}}{\sqrt{ab}}}}{\sqrt{ab}\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}d\mathit{\text{\_b}}+\mathit{\text{\_F1}}(\frac{a{z}^2-b{y}^2}{a},-\frac{-\sqrt{ab}x+\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right)}{\sqrt{ab}})){\mathrm{e}}^{{\int }_{}^y-\frac{(-\ln \left(\frac{\mathit{\text{\_a}}ab+\sqrt{(a{z}^2+({\mathit{\text{\_a}}}^2-y^2)b)a}\sqrt{ab}}{\sqrt{ab}}\right)+\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right))c+(-cx-k)\sqrt{ab}}{\sqrt{ab}\sqrt{(a{z}^2+({\mathit{\text{\_a}}}^2-y^2)b)a}}d\mathit{\text{\_a}}}$$

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7.9.1.3 [1923] Problem 3

problem number 1923

Added Jan 6, 2020.

Problem Chapter 9.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+(a_{1}x+a_0)w_y+(b_{1}x+b_0)w_z=(c_{1}x+c_0)w+s_{1}x+s_0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x+a0)*D[w[x,y,z],y]+(b1*x+b0)*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]];
 

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

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*x+a__0)*diff(w(x,y,z),y)+ (b__1*x+b__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'));
 

$$w(x,y,z)=\frac{(2c_1^{\frac{5}{2}}\mathit{\text{\_F1}}(-\frac{1}{2}{a}_1{x}^2-a_{0}x+y,-\frac{1}{2}{b}_1{x}^2-b_{0}x+z)-2c_1^{\frac{3}{2}}{s}_1{\mathrm{e}}^{-\frac{1}{2}{c}_1{x}^2-c_{0}x}+\sqrt{2}\sqrt{\pi }(-c_0{s}_1+c_1{s}_0)c_1\mathrm{erf}\left(\frac{\sqrt{2}(\sqrt{c_1}x+\frac{c_0}{\sqrt{c_1}})}{2}\right){\mathrm{e}}^{\frac{c_0^2}{2c_1}}){\mathrm{e}}^{\frac{(c_{1}x+2c_0)x}{2}}}{2c_1^{\frac{5}{2}}}$$

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7.9.1.4 [1924] Problem 4

problem number 1924

Added Jan 6, 2020.

Problem Chapter 9.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+(b_{1}x+b_0)w_y+(c_{1}y+c_0)w_z=aw+s_{1}x+s_0$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to -\frac{a^2(-e^{ax})c_1(-\text{b0}x-\frac{\text{b1}{x}^2}{2}+y,\frac{1}{2}\text{b0}\text{c1}{x}^2+\frac{1}{3}\text{b1}\text{c1}{x}^3-\text{c0}x-\text{c1}xy+z)+a\text{s0}+a\text{s1}x+\text{s1}}{a^2}\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (b__1*x+b__0)*diff(w(x,y,z),y)+ (c__1*x+c__0)*diff(w(x,y,z),z)=a*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'));
 

$$w(x,y,z)=\frac{a^2\mathit{\text{\_F1}}(-\frac{1}{2}{b}_1{x}^2-b_{0}x+y,-\frac{1}{2}{c}_1{x}^2-c_{0}x+z){\mathrm{e}}^{ax}+(-s_{1}x-s_0)a-s_1}{a^2}$$

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7.9.1.5 [1925] Problem 5

problem number 1925

Added Jan 6, 2020.

Problem Chapter 9.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a*y+k1*x+k0)*D[w[x,y,z],y]+(b*z+n1*x+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]];
 

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

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a*y+k__1*x+k__0)*diff(w(x,y,z),y)+ (b*z+n__1*x+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'));
 

$$w(x,y,z)=\frac{(2c_1^{\frac{5}{2}}\mathit{\text{\_F1}}(\frac{(a^{2}y+(k_{1}x+k_0)a+k_1){\mathrm{e}}^{-ax}}{a^2},\frac{(b^{2}z+(n_{1}x+n_0)b+n_1){\mathrm{e}}^{-bx}}{b^2})-2c_1^{\frac{3}{2}}{s}_1{\mathrm{e}}^{-\frac{1}{2}{c}_1{x}^2-c_{0}x}+\sqrt{2}\sqrt{\pi }(-c_0{s}_1+c_1{s}_0)c_1\mathrm{erf}\left(\frac{\sqrt{2}(\sqrt{c_1}x+\frac{c_0}{\sqrt{c_1}})}{2}\right){\mathrm{e}}^{\frac{c_0^2}{2c_1}}){\mathrm{e}}^{\frac{(c_{1}x+2c_0)x}{2}}}{2c_1^{\frac{5}{2}}}$$

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7.9.1.6 [1926] Problem 6

problem number 1926

Added Jan 6, 2020.

Problem Chapter 9.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+(a_{2}y+a_{1}x+a_0)w_y+(b_{3}z+b_{2}y+b_{1}x+b_0)w_z=(c_{3}z+c_{2}y+c_{1}x+c_0)w+s_{3}z+s_{2}y+s_{1}x+s_0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a2*y+a1*x+a0)*D[w[x,y,z],y]+(b3*z+b2*y+b1*x+b0)*D[w[x,y,z],z]==(c3*z+c2*y+c1*x+c0)*w[x,y,z]+s3*z+s2*y+s1*x+s0; 
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*y+a__1*x+a__0)*diff(w(x,y,z),y)+ (b__3*z+b__2*y+b__1*x+b__0)*diff(w(x,y,z),z)=(c__3*z+c__2*y+c__1*x+c__0)*w(x,y,z)+s__3*z+s__2*y+s__1*x+s__0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

time expired

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7.9.1.7 [1927] Problem 7

problem number 1927

Added Jan 6, 2020.

Problem Chapter 9.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$ax{w}_x+bx{w}_y+cz{w}_z=(\alpha x+\beta )w+px+q$$

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

$$\left\{\{w(x,y,z)\to \frac{e^{\frac{\alpha x}{a}}(-{\left(\frac{\alpha x}{a}\right)}^{\frac{\beta }{a}}(ap\mathrm{Gamma}(1-\frac{\beta }{a},\frac{\alpha x}{a})+\alpha q\mathrm{Gamma}(-\frac{\beta }{a},\frac{\alpha x}{a}))+a\alpha x^{\frac{\beta }{a}}{c}_1(y-\frac{bx}{a},z{x}^{-\frac{c}{a}}))}{a\alpha }\}\right\}$$

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

$$w(x,y,z)=-\frac{3((\alpha x+a-\beta )(a-\frac{\beta }{3})a^{2}q{x}^{-\frac{\beta }{a}-1}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{-\frac{a-\beta }{2a}}\mathrm{WhittakerM}(\frac{-a-\beta }{2a},\frac{2a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}-\frac{4{(a-\frac{\beta }{2})}^{2}a\beta p{x}^{-\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{\frac{-2a+\beta }{2a}}\mathrm{WhittakerM}(\frac{2a-\beta }{2a},\frac{3a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}}{3}+{(a-\beta )}^2(a-\frac{\beta }{3})aq{x}^{-\frac{\beta }{a}-1}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{-\frac{a-\beta }{2a}}\mathrm{WhittakerM}(\frac{a-\beta }{2a},\frac{2a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}-2(\frac{(\frac{\alpha x}{2}+a-\frac{\beta }{2})a^{2}p{x}^{-\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{\frac{-2a+\beta }{2a}}\mathrm{WhittakerM}(-\frac{\beta }{2a},\frac{3a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}}{3}+(a-\frac{\beta }{2})(a-\beta )(a-\frac{\beta }{3})\alpha \mathit{\text{\_F1}}(\frac{ay-bx}{a},z{x}^{-\frac{c}{a}}))\beta )x^{\frac{\beta }{a}}{\mathrm{e}}^{\frac{\alpha x}{a}}}{(a-\beta )(2a-\beta )(3a-\beta )\alpha \beta }$$

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7.9.1.8 [1928] Problem 8

problem number 1928

Added Jan 6, 2020.

Problem Chapter 9.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$ax{w}_x+by{w}_y+cz{w}_z=(\alpha x+\beta )w+px+q$$

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

$$\left\{\{w(x,y,z)\to \frac{e^{\frac{\alpha x}{a}}(-{\left(\frac{\alpha x}{a}\right)}^{\frac{\beta }{a}}(ap\mathrm{Gamma}(1-\frac{\beta }{a},\frac{\alpha x}{a})+\alpha q\mathrm{Gamma}(-\frac{\beta }{a},\frac{\alpha x}{a}))+a\alpha x^{\frac{\beta }{a}}{c}_1(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}}))}{a\alpha }\}\right\}$$

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

$$w(x,y,z)=-\frac{3((\alpha x+a-\beta )(a-\frac{\beta }{3})a^{2}q{x}^{-\frac{\beta }{a}-1}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{-\frac{a-\beta }{2a}}\mathrm{WhittakerM}(\frac{-a-\beta }{2a},\frac{2a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}-\frac{4{(a-\frac{\beta }{2})}^{2}a\beta p{x}^{-\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{\frac{-2a+\beta }{2a}}\mathrm{WhittakerM}(\frac{2a-\beta }{2a},\frac{3a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}}{3}+{(a-\beta )}^2(a-\frac{\beta }{3})aq{x}^{-\frac{\beta }{a}-1}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{-\frac{a-\beta }{2a}}\mathrm{WhittakerM}(\frac{a-\beta }{2a},\frac{2a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}-2(\frac{(\frac{\alpha x}{2}+a-\frac{\beta }{2})a^{2}p{x}^{-\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{\frac{\beta }{a}}{\left(\frac{\alpha }{a}\right)}^{-\frac{\beta }{a}}{\left(\frac{\alpha x}{a}\right)}^{\frac{-2a+\beta }{2a}}\mathrm{WhittakerM}(-\frac{\beta }{2a},\frac{3a-\beta }{2a},\frac{\alpha x}{a}){\mathrm{e}}^{-\frac{\alpha x}{2a}}}{3}+(a-\frac{\beta }{2})(a-\beta )(a-\frac{\beta }{3})\alpha \mathit{\text{\_F1}}(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}}))\beta )x^{\frac{\beta }{a}}{\mathrm{e}}^{\frac{\alpha x}{a}}}{(a-\beta )(2a-\beta )(3a-\beta )\alpha \beta }$$

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7.9.1.9 [1929] Problem 9

problem number 1929

Added Jan 6, 2020.

Problem Chapter 9.2.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$x{w}_x+az{w}_y+by{w}_z=(cx+k)w+px+q$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to \frac{e^{cx}(-(cx{)}^k(p\mathrm{Gamma}(1-k,cx)+cq\mathrm{Gamma}(-k,cx))+c{x}^k{c}_1(iy\sinh (\sqrt{a}\sqrt{b}\log (x))-\frac{i\sqrt{a}z\cosh (\sqrt{a}\sqrt{b}\log (x))}{\sqrt{b}},y\cosh (\sqrt{a}\sqrt{b}\log (x))-\frac{\sqrt{a}z\sinh (\sqrt{a}\sqrt{b}\log (x))}{\sqrt{b}}))}{c}\}\right\}$$

Maple ✓

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

$$w(x,y,z)=({\int }_{}^y\frac{(px{\left(\frac{\mathit{\text{\_b}}ab+\sqrt{ab}\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}{\sqrt{ab}}\right)}^{\frac{1}{\sqrt{ab}}}{(az+\sqrt{ab}y)}^{-\frac{\sqrt{ab}}{ab}}+q){\mathrm{e}}^{-(\int \frac{cx{\left(\frac{\mathit{\text{\_b}}ab+\sqrt{ab}\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}{\sqrt{ab}}\right)}^{\frac{1}{\sqrt{ab}}}{(az+\sqrt{ab}y)}^{-\frac{\sqrt{ab}}{ab}}+k}{\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}d\mathit{\text{\_b}})}}{\sqrt{(a{z}^2+({\mathit{\text{\_b}}}^2-y^2)b)a}}d\mathit{\text{\_b}}+\mathit{\text{\_F1}}(\frac{a{z}^2-b{y}^2}{a},x{(az+\sqrt{ab}y)}^{-\frac{\sqrt{ab}}{ab}})){\mathrm{e}}^{{\int }_{}^y\frac{cx{\left(\frac{\mathit{\text{\_a}}ab+\sqrt{(a{z}^2+({\mathit{\text{\_a}}}^2-y^2)b)a}\sqrt{ab}}{\sqrt{ab}}\right)}^{\frac{1}{\sqrt{ab}}}{(az+\sqrt{ab}y)}^{-\frac{\sqrt{ab}}{ab}}+k}{\sqrt{(a{z}^2+({\mathit{\text{\_a}}}^2-y^2)b)a}}d\mathit{\text{\_a}}}$$

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