6.9.4 2.4

6.9.4.1 [1941] Problem 1

problem number 1941

Added Jan 16, 2020.

Problem Chapter 9.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{w}_x+b{w}_y+c{w}_z=k{x}^{n}w+s{x}^m$$

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]==k*x^n*w[x,y,z]+ s*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{k{x}^{n+1}}{an+a}}(c_1(y-\frac{bx}{a},z-\frac{cx}{a})-\frac{s{x}^{m+1}{\left(\frac{k{x}^{n+1}}{an+a}\right)}^{-\frac{m+1}{n+1}}\text{Gamma}(\frac{m+1}{n+1},\frac{k{x}^{n+1}}{an+a})}{a(n+1)})\}\right\}$$

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)=k*x^n*w(x,y,z)+ s*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\frac{1}{ak(m+2n+3)(m+n+2)(m+1)}{\mathrm{e}}^{\frac{x^{n+1}k}{(n+1)a}}({\left(\frac{k}{(n+1)a}\right)}^{\frac{-m-1}{n+1}}s(x^{n+1}k+a(m+n+2)){\left(\frac{x^{n+1}k}{(n+1)a}\right)}^{\frac{-m-n-2}{2n+2}}{\left(\frac{k}{(n+1)a}\right)}^{\frac{m+1}{n+1}}{\mathrm{e}}^{-1/2\frac{x^{n+1}k}{(n+1)a}}{(n+1)}^2{x}^{-n+m}\mathrm{WhittakerM}(\frac{-n+m}{2n+2},\frac{m+2n+3}{2n+2},\frac{x^{n+1}k}{(n+1)a})+({\mathrm{e}}^{-1/2\frac{x^{n+1}k}{(n+1)a}}{\left(\frac{k}{(n+1)a}\right)}^{\frac{-m-1}{n+1}}{\left(\frac{k}{(n+1)a}\right)}^{\frac{m+1}{n+1}}{\left(\frac{x^{n+1}k}{(n+1)a}\right)}^{\frac{-m-n-2}{2n+2}}{x}^{-n+m}s(n+1)(m+n+2)\mathrm{WhittakerM}(\frac{m+n+2}{2n+2},\frac{m+2n+3}{2n+2},\frac{x^{n+1}k}{(n+1)a})+\mathit{\_}\mathit{F}\mathit{1}(\frac{ay-bx}{a},\frac{za-cx}{a})k(m+1)(m+2n+3))a(m+n+2))$$

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6.9.4.2 [1942] Problem 2

problem number 1942

Added Jan 16, 2020.

Problem Chapter 9.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{w}_x+by{w}_y+cz{w}_z=k{x}^{n}w+s{x}^m$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to e^{\frac{k{x}^{n+1}}{an+a}}(c_1(y{e}^{-\frac{bx}{a}},z{e}^{-\frac{cx}{a}})-\frac{s{x}^{m+1}{\left(\frac{k{x}^{n+1}}{an+a}\right)}^{-\frac{m+1}{n+1}}\text{Gamma}(\frac{m+1}{n+1},\frac{k{x}^{n+1}}{an+a})}{a(n+1)})\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{1}{ak(m+2n+3)(m+n+2)(m+1)}{\mathrm{e}}^{\frac{x^{n+1}k}{(n+1)a}}({\left(\frac{k}{(n+1)a}\right)}^{\frac{-m-1}{n+1}}s(x^{n+1}k+a(m+n+2)){\left(\frac{x^{n+1}k}{(n+1)a}\right)}^{\frac{-m-n-2}{2n+2}}{\left(\frac{k}{(n+1)a}\right)}^{\frac{m+1}{n+1}}{\mathrm{e}}^{-1/2\frac{x^{n+1}k}{(n+1)a}}{(n+1)}^2{x}^{-n+m}\mathrm{WhittakerM}(\frac{-n+m}{2n+2},\frac{m+2n+3}{2n+2},\frac{x^{n+1}k}{(n+1)a})+a({\mathrm{e}}^{-1/2\frac{x^{n+1}k}{(n+1)a}}{\left(\frac{k}{(n+1)a}\right)}^{\frac{-m-1}{n+1}}{\left(\frac{k}{(n+1)a}\right)}^{\frac{m+1}{n+1}}{\left(\frac{x^{n+1}k}{(n+1)a}\right)}^{\frac{-m-n-2}{2n+2}}{x}^{-n+m}s(n+1)(m+n+2)\mathrm{WhittakerM}(\frac{m+n+2}{2n+2},\frac{m+2n+3}{2n+2},\frac{x^{n+1}k}{(n+1)a})+\mathit{\_}\mathit{F}\mathit{1}(y{\mathrm{e}}^{-\frac{bx}{a}},z{\mathrm{e}}^{-\frac{cx}{a}})k(m+1)(m+2n+3))(m+n+2))$$

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6.9.4.3 [1943] Problem 3

problem number 1943

Added Jan 16, 2020.

Problem Chapter 9.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+az{w}_y+by{w}_z=c{x}^{n}w+s{x}^m$$

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^n*w[x,y,z]+ s*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{c{x}^{n+1}}{n+1}}(c_1(\frac{e^{-\sqrt{a}\sqrt{b}x}(\sqrt{b}y(e^{2\sqrt{a}\sqrt{b}x}+1)-\sqrt{a}z(e^{2\sqrt{a}\sqrt{b}x}-1))}{2\sqrt{b}},\frac{e^{-\sqrt{a}\sqrt{b}x}(\sqrt{a}z(e^{2\sqrt{a}\sqrt{b}x}+1)-\sqrt{b}y(e^{2\sqrt{a}\sqrt{b}x}-1))}{2\sqrt{a}})-\frac{s{x}^{m+1}{\left(\frac{c{x}^{n+1}}{n+1}\right)}^{-\frac{m+1}{n+1}}\text{Gamma}(\frac{m+1}{n+1},\frac{c{x}^{n+1}}{n+1})}{n+1})\}\right\}$$

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^n*w(x,y,z)+ s*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=({\int }^y\frac{s}{\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}}{\left(\frac{1}{\sqrt{ab}}(x\sqrt{ab}-\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right)+\ln \left(\frac{ab\mathit{\_}\mathit{b}+\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}\sqrt{ab}}{\sqrt{ab}}\right))\right)}^m{\mathrm{e}}^{-c\int \frac{1}{\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}}{\left(\frac{1}{\sqrt{ab}}(x\sqrt{ab}-\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right)+\ln \left(\frac{ab\mathit{\_}\mathit{b}+\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}\sqrt{ab}}{\sqrt{ab}}\right))\right)}^n\mathrm{d}\mathit{\_}\mathit{b}}d\mathit{\_}\mathit{b}+\mathit{\_}\mathit{F}\mathit{1}(\frac{a{z}^2-b{y}^2}{a},-\frac{1}{\sqrt{ab}}(-x\sqrt{ab}+\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right)))){\mathrm{e}}^{{\int }^y\frac{c}{\sqrt{a(a{z}^2+({\mathit{\_}\mathit{a}}^2-y^2)b)}}{\left(\frac{1}{\sqrt{ab}}(x\sqrt{ab}-\ln \left(\frac{aby+\sqrt{a^2{z}^2}\sqrt{ab}}{\sqrt{ab}}\right)+\ln \left(\frac{ab\mathit{\_}\mathit{a}+\sqrt{a(a{z}^2+({\mathit{\_}\mathit{a}}^2-y^2)b)}\sqrt{ab}}{\sqrt{ab}}\right))\right)}^{n}d\mathit{\_}\mathit{a}}$$

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6.9.4.4 [1944] Problem 4

problem number 1944

Added Jan 16, 2020.

Problem Chapter 9.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+a{x}^n{w}_y+b{x}^m{w}_z=c{x}^{k}w+s{x}^r$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to e^{\frac{c{x}^{k+1}}{k+1}}(c_1(\frac{-a{x}^{n+1}+ny+y}{n+1},\frac{-b{x}^{m+1}+mz+z}{m+1})-\frac{s{x}^{r+1}{\left(\frac{c{x}^{k+1}}{k+1}\right)}^{-\frac{r+1}{k+1}}\text{Gamma}(\frac{r+1}{k+1},\frac{c{x}^{k+1}}{k+1})}{k+1})\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{1}{c(2k+r+3)(r+1)(k+r+2)}(2(k+r/2+3/2)(r+1)c(k+r+2)\mathit{\_}\mathit{F}\mathit{1}(\frac{-xa{x}^n+y(n+1)}{n+1},\frac{-xb{x}^m+z(m+1)}{m+1})+{\mathrm{e}}^{-\frac{x^{k+1}c}{2k+2}}{x}^{-k+r}(k+1){\left(\frac{x^{k+1}c}{k+1}\right)}^{\frac{-k-r-2}{2k+2}}s{\left(\frac{c}{k+1}\right)}^{\frac{-r-1}{k+1}}((k+1)(x^{k+1}c+k+r+2)\mathrm{WhittakerM}(\frac{-k+r}{2k+2},\frac{2k+r+3}{2k+2},\frac{x^{k+1}c}{k+1})+\mathrm{WhittakerM}(\frac{k+r+2}{2k+2},\frac{2k+r+3}{2k+2},\frac{x^{k+1}c}{k+1}){(k+r+2)}^2){\left(\frac{c}{k+1}\right)}^{\frac{r+1}{k+1}}){\mathrm{e}}^{\frac{x^{k+1}c}{k+1}}$$

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6.9.4.5 [1945] Problem 5

problem number 1945

Added Jan 16, 2020.

Problem Chapter 9.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+b{x}^n{w}_y+c{y}^m{w}_z=aw+s{x}^k$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to e^{ax}(c_1(\frac{-b{x}^{n+1}+ny+y}{n+1},\frac{-c{x}^{m+1}+mz+z}{m+1})-\frac{s{x}^k(ax{)}^{-k}\text{Gamma}(k+1,ax)}{a})\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{{\mathrm{e}}^{ax}}{a(k+1)}(a(k+1)\mathit{\_}\mathit{F}\mathit{1}(\frac{-xb{x}^n+y(n+1)}{n+1},\frac{-xc{x}^m+z(m+1)}{m+1})+s{x}^k{\left(ax\right)}^{-k/2}{\mathrm{e}}^{-1/2ax}\mathrm{WhittakerM}(k/2,k/2+1/2,ax))$$

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6.9.4.6 [1946] Problem 6

problem number 1946

Added Jan 16, 2020.

Problem Chapter 9.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+(ay+\beta x^n)w_y+(bz+\gamma x^m)w_z=c{x}^{k}w+s{x}^r$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to e^{\frac{c{x}^{k+1}}{k+1}}(c_1(\gamma b^{-m-1}\text{Gamma}(m+1,bx)+z{e}^{-bx},\beta a^{-n-1}\text{Gamma}(n+1,ax)+y{e}^{-ax})-\frac{s{x}^{r+1}{\left(\frac{c{x}^{k+1}}{k+1}\right)}^{-\frac{r+1}{k+1}}\text{Gamma}(\frac{r+1}{k+1},\frac{c{x}^{k+1}}{k+1})}{k+1})\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{1}{c(2k+r+3)(r+1)(k+r+2)}(2(k+r/2+3/2)(r+1)c(k+r+2)\mathit{\_}\mathit{F}\mathit{1}(\frac{(-x^n{\left(ax\right)}^{-n/2}\mathrm{WhittakerM}(n/2,n/2+1/2,ax){\mathrm{e}}^{1/2ax}\beta +ay(n+1)){\mathrm{e}}^{-ax}}{(n+1)a},\frac{(-x^m{\mathrm{e}}^{1/2bx}{\left(bx\right)}^{-m/2}\mathrm{WhittakerM}(m/2,m/2+1/2,bx)\gamma +bz(m+1)){\mathrm{e}}^{-bx}}{b(m+1)})+{\mathrm{e}}^{-\frac{x^{k+1}c}{2k+2}}{x}^{-k+r}(k+1){\left(\frac{x^{k+1}c}{k+1}\right)}^{\frac{-k-r-2}{2k+2}}s{\left(\frac{c}{k+1}\right)}^{\frac{-r-1}{k+1}}((k+1)(x^{k+1}c+k+r+2)\mathrm{WhittakerM}(\frac{-k+r}{2k+2},\frac{2k+r+3}{2k+2},\frac{x^{k+1}c}{k+1})+\mathrm{WhittakerM}(\frac{k+r+2}{2k+2},\frac{2k+r+3}{2k+2},\frac{x^{k+1}c}{k+1}){(k+r+2)}^2){\left(\frac{c}{k+1}\right)}^{\frac{r+1}{k+1}}){\mathrm{e}}^{\frac{x^{k+1}c}{k+1}}$$

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6.9.4.7 [1947] Problem 7

problem number 1947

Added Jan 16, 2020.

Problem Chapter 9.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+(a_1{x}^{n_1}y+a_2{x}^{n_2})w_y+(b_1{y}^{m_1}z+b_2{y}^{m_2})w_z=cw+s_{1}x{y}^{k_1}+s_2{x}^{k_2}z$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x^n1*y + a2*x^n2)*D[w[x,y,z],y]+(b1*y^m1*z + b2*y^m2)*D[w[x,y,z],z]==c*w[x,y,z]+ s1*x*y^k1+ s2*x^k2*z; 
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{a2}(\text{n1}+1{)}^{\frac{\text{n2}-\text{n1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}){\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}+e^{-\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}y,e^{-{\int }_1^x\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}z-{\int }_1^x\text{b2}{e}^{-{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^x\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}\right),\{K[1],1,x\}]dK[1]}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[2{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[2{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m2}}dK[2])+{\int }_1^x{e}^{-cK[3]}(\text{s1}K[3]{\left(\frac{{\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[3{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}((\text{n1}+1)y{\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}+\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}+1}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}+1}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[3{]}^{\text{n1}+1}}{\text{n1}+1}))}{\text{n1}+1}\right)}^{\text{k1}}+e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[3]}\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1],\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]-{\int }_1^x\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}\text{s2}zK[3{]}^{\text{k2}}-e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[3]}\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1],\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\text{s2}K[3{]}^{\text{k2}}{\int }_1^x\text{b2}{e}^{-{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^x\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}\right),\{K[1],1,x\}]dK[1]}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[2{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[2{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m2}}dK[2]+e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[3]}\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1],\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\text{s2}K[3{]}^{\text{k2}}{\int }_1^{K[3]}\text{b2}{e}^{-{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{\text{InverseFunction}[\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2],1,2][\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[2]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log \left(e^{{\int }_1^{K[3]}\text{b1}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[1{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[1{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m1}}dK[1]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}\right),\{K[1],1,K[2]\}],\{K[1],1,K[3]\}]}\right),\{K[1],1,K[3]\}]dK[1]}{({\text{a1}}^{-\frac{\text{n2}+1}{\text{n1}+1}}{e}^{\frac{\text{a1}(K[2{]}^{\text{n1}+1}-x^{\text{n1}+1})}{\text{n1}+1}}(\text{n1}+1{)}^{-\frac{\text{n1}}{\text{n1}+1}-1}(\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1})(\text{n1}+1{)}^{\frac{\text{n1}+\text{n2}+1}{\text{n1}+1}}+({\text{a1}}^{\frac{\text{n2}+1}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n1}}{\text{n1}+1}}y-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{n1}+1}}{\text{n1}+1}}(\text{n1}+1{)}^{\frac{\text{n2}}{\text{n1}+1}}\text{Gamma}(\frac{\text{n2}+1}{\text{n1}+1},\frac{\text{a1}K[2{]}^{\text{n1}+1}}{\text{n1}+1}))(\text{n1}+1)))}^{\text{m2}}dK[2])dK[3])\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*x^(n__1)*y + a__2*x^(n__2))*diff(w(x,y,z),y)+ (b__1*y^(m__1)*z + b__2*y^(m__2))*diff(w(x,y,z),z)=c*w(x,y,z)+ s__1*x*y^(k__1)+ s__2*x^(k__2)*z; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

____________________________________________________________________________________

6.9.4.8 [1948] Problem 8

problem number 1948

Added Jan 16, 2020.

Problem Chapter 9.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+(a_1{x}^{{\lambda }_1}y+a_2{x}^{{\lambda }_2}{y}^k)w_y+(b_1{x}^{{\beta }_1}z+b_2{x}^{{\beta }_2}{z}^m)w_z=c_1{x}^{{\gamma }_1}w+c_2{y}^{{\gamma }_2}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x^lambda1*y + a2*x^lambda2*y^k)*D[w[x,y,z],y]+(b1*x^beta1*z + b2*x^beta2*z^m)*D[w[x,y,z],z]==c1*x^gamma1*w[x,y,z]+ c2*y^gamma2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{\text{c1}{x}^{\text{gamma1}+1}}{\text{gamma1}+1}}({\int }_1^x\text{c2}{e}^{-\frac{\text{c1}K[1{]}^{\text{gamma1}+1}}{\text{gamma1}+1}}{\left({\left(\frac{(-1{)}^{-\frac{\text{lambda2}+1}{\text{lambda1}+1}}{\text{a1}}^{-\frac{\text{lambda2}+1}{\text{lambda1}+1}}\exp (-\frac{\text{a1}(x^{\text{lambda1}+1}+(k-1)K[1{]}^{\text{lambda1}+1})}{\text{lambda1}+1})(k-1{)}^{-\frac{\text{lambda2}+1}{\text{lambda1}+1}}{y}^{-k}(-\text{a2}{e}^{\frac{\text{a1}{x}^{\text{lambda1}+1}}{\text{lambda1}+1}}(k-1)(\text{lambda1}+1{)}^{\frac{\text{lambda2}+1}{\text{lambda1}+1}}\text{Gamma}(\frac{\text{lambda2}+1}{\text{lambda1}+1},-\frac{\text{a1}(k-1)x^{\text{lambda1}+1}}{\text{lambda1}+1})y^k+\text{a2}{e}^{\frac{\text{a1}{x}^{\text{lambda1}+1}}{\text{lambda1}+1}}(k-1)(\text{lambda1}+1{)}^{\frac{\text{lambda2}+1}{\text{lambda1}+1}}\text{Gamma}(\frac{\text{lambda2}+1}{\text{lambda1}+1},-\frac{\text{a1}(k-1)K[1{]}^{\text{lambda1}+1}}{\text{lambda1}+1})y^k+(-1{)}^{\frac{\text{lambda2}+1}{\text{lambda1}+1}}{\text{a1}}^{\frac{\text{lambda2}+1}{\text{lambda1}+1}}{e}^{\frac{\text{a1}k{x}^{\text{lambda1}+1}}{\text{lambda1}+1}}(k-1{)}^{\frac{\text{lambda2}+1}{\text{lambda1}+1}}(\text{lambda1}+1)y)}{\text{lambda1}+1}\right)}^{\frac{1}{1-k}}\right)}^{\text{gamma2}}dK[1]+c_1(\text{b2}(-1{)}^{\frac{\text{beta1}-\text{beta2}}{\text{beta1}+1}}(\text{beta1}+1{)}^{\frac{\text{beta2}-\text{beta1}}{\text{beta1}+1}}{\text{b1}}^{-\frac{\text{beta2}+1}{\text{beta1}+1}}(m-1{)}^{\frac{\text{beta1}-\text{beta2}}{\text{beta1}+1}}\text{Gamma}(\frac{\text{beta2}+1}{\text{beta1}+1},-\frac{\text{b1}(m-1)x^{\text{beta1}+1}}{\text{beta1}+1})+z^{1-m}{e}^{\frac{\text{b1}(m-1)x^{\text{beta1}+1}}{\text{beta1}+1}},\text{a2}(-1{)}^{\frac{\text{lambda1}-\text{lambda2}}{\text{lambda1}+1}}(\text{lambda1}+1{)}^{\frac{\text{lambda2}-\text{lambda1}}{\text{lambda1}+1}}{\text{a1}}^{-\frac{\text{lambda2}+1}{\text{lambda1}+1}}(k-1{)}^{\frac{\text{lambda1}-\text{lambda2}}{\text{lambda1}+1}}\text{Gamma}(\frac{\text{lambda2}+1}{\text{lambda1}+1},-\frac{\text{a1}(k-1)x^{\text{lambda1}+1}}{\text{lambda1}+1})+y^{1-k}{e}^{\frac{\text{a1}(k-1)x^{\text{lambda1}+1}}{\text{lambda1}+1}}))\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*x^(lambda__1)*y + a__2*x^(lambda__2)*y^k)*diff(w(x,y,z),y)+ (b__1*x^(beta__1)*z + b__2*x^(beta__2)*z^m)*diff(w(x,y,z),z)=c__1*x^(gamma__1)*w(x,y,z)+ c__2*y^(gamma__2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=({\int }^x{c}_2{\left({\left(\frac{1}{a_1(2{\lambda }_1+{\lambda }_2+3)({\lambda }_1+{\lambda }_2+2)({\lambda }_2+1)}(((-{\lambda }_1-{\lambda }_2-2)x^{{\lambda }_2-{\lambda }_1}+a_1{x}^{{\lambda }_2+1}(k-1))a_2{y}^{\frac{k{\lambda }_1}{{\lambda }_1+1}}{y}^{\frac{k}{{\lambda }_1+1}}{\mathrm{e}}^{\frac{a_1(k-1)x^{{\lambda }_1+1}}{2{\lambda }_1+2}}{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}{({\lambda }_1+1)}^2{(-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})}^{\frac{-{\lambda }_1-{\lambda }_2-2}{2{\lambda }_1+2}}\mathrm{WhittakerM}(\frac{{\lambda }_2-{\lambda }_1}{2{\lambda }_1+2},\frac{2{\lambda }_1+{\lambda }_2+3}{2{\lambda }_1+2},-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})-((-{\lambda }_1-{\lambda }_2-2){\mathit{\_}\mathit{a}}^{{\lambda }_2-{\lambda }_1}+{\mathit{\_}\mathit{a}}^{{\lambda }_2+1}{a}_1(k-1)){({\lambda }_1+1)}^2{\mathrm{e}}^{\frac{a_1{\mathit{\_}\mathit{a}}^{{\lambda }_1+1}(k-1)}{2{\lambda }_1+2}}{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}{(-\frac{a_1{\mathit{\_}\mathit{a}}^{{\lambda }_1+1}(k-1)}{{\lambda }_1+1})}^{\frac{-{\lambda }_1-{\lambda }_2-2}{2{\lambda }_1+2}}{a}_2{y}^{\frac{k}{{\lambda }_1+1}}{y}^{\frac{k{\lambda }_1}{{\lambda }_1+1}}\mathrm{WhittakerM}(\frac{{\lambda }_2-{\lambda }_1}{2{\lambda }_1+2},\frac{2{\lambda }_1+{\lambda }_2+3}{2{\lambda }_1+2},-\frac{a_1{\mathit{\_}\mathit{a}}^{{\lambda }_1+1}(k-1)}{{\lambda }_1+1})+2(-1/2{\mathrm{e}}^{\frac{a_1(k-1)x^{{\lambda }_1+1}}{2{\lambda }_1+2}}{x}^{{\lambda }_2-{\lambda }_1}{(-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})}^{\frac{-{\lambda }_1-{\lambda }_2-2}{2{\lambda }_1+2}}{a}_2{y}^{\frac{k{\lambda }_1}{{\lambda }_1+1}}{y}^{\frac{k}{{\lambda }_1+1}}{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}({\lambda }_1+1)({\lambda }_1+{\lambda }_2+2)\mathrm{WhittakerM}(\frac{{\lambda }_1+{\lambda }_2+2}{2{\lambda }_1+2},\frac{2{\lambda }_1+{\lambda }_2+3}{2{\lambda }_1+2},-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})+1/2{\mathrm{e}}^{\frac{a_1{\mathit{\_}\mathit{a}}^{{\lambda }_1+1}(k-1)}{2{\lambda }_1+2}}{\mathit{\_}\mathit{a}}^{{\lambda }_2-{\lambda }_1}{(-\frac{a_1{\mathit{\_}\mathit{a}}^{{\lambda }_1+1}(k-1)}{{\lambda }_1+1})}^{\frac{-{\lambda }_1-{\lambda }_2-2}{2{\lambda }_1+2}}{a}_2{y}^{\frac{k{\lambda }_1}{{\lambda }_1+1}}{y}^{\frac{k}{{\lambda }_1+1}}{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}({\lambda }_1+1)({\lambda }_1+{\lambda }_2+2)\mathrm{WhittakerM}(\frac{{\lambda }_1+{\lambda }_2+2}{2{\lambda }_1+2},\frac{2{\lambda }_1+{\lambda }_2+3}{2{\lambda }_1+2},-\frac{a_1{\mathit{\_}\mathit{a}}^{{\lambda }_1+1}(k-1)}{{\lambda }_1+1})+{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}k}{{\lambda }_1+1}}({\lambda }_2+1)y^{{({\lambda }_1+1)}^{-1}}{a}_1({\lambda }_1+{\lambda }_2/2+3/2)y^{\frac{{\lambda }_1}{{\lambda }_1+1}})({\lambda }_1+{\lambda }_2+2)){\left({\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}\right)}^{-1}{\left(y^{\frac{k}{{\lambda }_1+1}}\right)}^{-1}{\left(y^{\frac{k{\lambda }_1}{{\lambda }_1+1}}\right)}^{-1}\right)}^{-{(k-1)}^{-1}}{\mathrm{e}}^{\frac{a_1{\mathit{\_}\mathit{a}}^{{\lambda }_1+1}}{{\lambda }_1+1}}\right)}^{{\gamma }_2}{\mathrm{e}}^{-\frac{c_1{\mathit{\_}\mathit{a}}^{{\gamma }_1+1}}{{\gamma }_1+1}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}(\frac{1}{a_1(2{\lambda }_1+{\lambda }_2+3)({\lambda }_1+{\lambda }_2+2)({\lambda }_2+1)}(((-{\lambda }_1-{\lambda }_2-2)x^{{\lambda }_2-{\lambda }_1}+a_1{x}^{{\lambda }_2+1}(k-1))a_2{y}^{\frac{k{\lambda }_1}{{\lambda }_1+1}}{y}^{\frac{k}{{\lambda }_1+1}}{\mathrm{e}}^{\frac{a_1(k-1)x^{{\lambda }_1+1}}{2{\lambda }_1+2}}{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}{({\lambda }_1+1)}^2{(-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})}^{\frac{-{\lambda }_1-{\lambda }_2-2}{2{\lambda }_1+2}}\mathrm{WhittakerM}(\frac{{\lambda }_2-{\lambda }_1}{2{\lambda }_1+2},\frac{2{\lambda }_1+{\lambda }_2+3}{2{\lambda }_1+2},-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})+2({\lambda }_1+{\lambda }_2+2)(-1/2{\mathrm{e}}^{\frac{a_1(k-1)x^{{\lambda }_1+1}}{2{\lambda }_1+2}}{x}^{{\lambda }_2-{\lambda }_1}{(-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})}^{\frac{-{\lambda }_1-{\lambda }_2-2}{2{\lambda }_1+2}}{a}_2{y}^{\frac{k{\lambda }_1}{{\lambda }_1+1}}{y}^{\frac{k}{{\lambda }_1+1}}{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}({\lambda }_1+1)({\lambda }_1+{\lambda }_2+2)\mathrm{WhittakerM}(\frac{{\lambda }_1+{\lambda }_2+2}{2{\lambda }_1+2},\frac{2{\lambda }_1+{\lambda }_2+3}{2{\lambda }_1+2},-\frac{a_1(k-1)x^{{\lambda }_1+1}}{{\lambda }_1+1})+{\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}k}{{\lambda }_1+1}}({\lambda }_2+1)y^{{({\lambda }_1+1)}^{-1}}{a}_1({\lambda }_1+{\lambda }_2/2+3/2)y^{\frac{{\lambda }_1}{{\lambda }_1+1}})){\left({\mathrm{e}}^{\frac{a_1{x}^{{\lambda }_1+1}}{{\lambda }_1+1}}\right)}^{-1}{\left(y^{\frac{k}{{\lambda }_1+1}}\right)}^{-1}{\left(y^{\frac{k{\lambda }_1}{{\lambda }_1+1}}\right)}^{-1},\frac{1}{b_1({\beta }_2+1)({\beta }_1+{\beta }_2+2)(2{\beta }_1+{\beta }_2+3)}(((-{\beta }_1-{\beta }_2-2)x^{-{\beta }_1+{\beta }_2}+b_1{x}^{{\beta }_2+1}(m-1))b_2{z}^{\frac{{\beta }_{1}m}{{\beta }_1+1}}{z}^{\frac{m}{{\beta }_1+1}}{\mathrm{e}}^{\frac{x^{{\beta }_1+1}{b}_1(m-1)}{2{\beta }_1+2}}{\mathrm{e}}^{\frac{b_1{x}^{{\beta }_1+1}}{{\beta }_1+1}}{({\beta }_1+1)}^2{(-\frac{x^{{\beta }_1+1}{b}_1(m-1)}{{\beta }_1+1})}^{\frac{-{\beta }_1-{\beta }_2-2}{2{\beta }_1+2}}\mathrm{WhittakerM}(\frac{-{\beta }_1+{\beta }_2}{2{\beta }_1+2},\frac{2{\beta }_1+{\beta }_2+3}{2{\beta }_1+2},-\frac{x^{{\beta }_1+1}{b}_1(m-1)}{{\beta }_1+1})+2({\beta }_1+{\beta }_2+2)(-1/2{\mathrm{e}}^{\frac{x^{{\beta }_1+1}{b}_1(m-1)}{2{\beta }_1+2}}{x}^{-{\beta }_1+{\beta }_2}{(-\frac{x^{{\beta }_1+1}{b}_1(m-1)}{{\beta }_1+1})}^{\frac{-{\beta }_1-{\beta }_2-2}{2{\beta }_1+2}}{b}_2{z}^{\frac{{\beta }_{1}m}{{\beta }_1+1}}{z}^{\frac{m}{{\beta }_1+1}}{\mathrm{e}}^{\frac{b_1{x}^{{\beta }_1+1}}{{\beta }_1+1}}({\beta }_1+1)({\beta }_1+{\beta }_2+2)\mathrm{WhittakerM}(\frac{{\beta }_1+{\beta }_2+2}{2{\beta }_1+2},\frac{2{\beta }_1+{\beta }_2+3}{2{\beta }_1+2},-\frac{x^{{\beta }_1+1}{b}_1(m-1)}{{\beta }_1+1})+{\mathrm{e}}^{\frac{b_1{x}^{{\beta }_1+1}m}{{\beta }_1+1}}({\beta }_2+1)z^{{({\beta }_1+1)}^{-1}}{b}_1({\beta }_1+{\beta }_2/2+3/2)z^{\frac{{\beta }_1}{{\beta }_1+1}})){\left(z^{\frac{{\beta }_{1}m}{{\beta }_1+1}}\right)}^{-1}{\left(z^{\frac{m}{{\beta }_1+1}}\right)}^{-1}{\left({\mathrm{e}}^{\frac{b_1{x}^{{\beta }_1+1}}{{\beta }_1+1}}\right)}^{-1})){\mathrm{e}}^{\frac{x^{{\gamma }_1+1}{c}_1}{{\gamma }_1+1}}$$

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6.9.4.9 [1949] Problem 9

problem number 1949

Added Jan 16, 2020.

Problem Chapter 9.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+(a_1{x}^{{\lambda }_1}y+a_2{x}^{{\lambda }_2}{y}^k)w_y+(b_1{y}^{{\beta }_1}z+b_2{y}^{{\beta }_2}{z}^m)w_z=c_1{x}^{{\gamma }_1}w+c_2{z}^{{\gamma }_2}$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(a1*x^lambda1*y + a2*x^lambda2*y^k)*D[w[x,y,z],y]+(b1*y^beta1*z + b2*y^beta2*z^m)*D[w[x,y,z],z]==c1*x^gamma1*w[x,y,z]+ c2*z^gamma2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✗

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a__1*x^(lambda__1)*y + a__2*x^(lambda__2)*y^k)*diff(w(x,y,z),y)+ (b__1*y^(beta__1)*z + b__2*y^(beta__2)*z^m)*diff(w(x,y,z),z)=c__1*x^(gamma__1)*w(x,y,z)+ c__2*z^(gamma__2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

time expired

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6.9.4.10 [1950] Problem 10

problem number 1950

Added Jan 16, 2020.

Problem Chapter 9.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$x{w}_x+ay{w}_y+bz{w}_z=c{x}^{n}w+k{x}^m$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to e^{\frac{c{x}^n}{n}}(c_1(y{x}^{-a},z{x}^{-b})-\frac{k{x}^m{\left(\frac{c{x}^n}{n}\right)}^{-\frac{m}{n}}\text{Gamma}(\frac{m}{n},\frac{c{x}^n}{n})}{n})\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{1}{m(m+n)(m+2n)c}{\mathrm{e}}^{\frac{x^{n}c}{n}}(x^{-n+m}{\left(\frac{c}{n}\right)}^{\frac{m}{n}}{\left(\frac{c}{n}\right)}^{-\frac{m}{n}}{\mathrm{e}}^{-1/2\frac{x^{n}c}{n}}{\left(\frac{x^{n}c}{n}\right)}^{-1/2\frac{m+n}{n}}k{n}^2(x^{n}c+m+n)\mathrm{WhittakerM}(1/2\frac{-n+m}{n},1/2\frac{m+2n}{n},\frac{x^{n}c}{n})+(x^{-n+m}{\left(\frac{c}{n}\right)}^{\frac{m}{n}}{\left(\frac{c}{n}\right)}^{-\frac{m}{n}}{\mathrm{e}}^{-1/2\frac{x^{n}c}{n}}{\left(\frac{x^{n}c}{n}\right)}^{-1/2\frac{m+n}{n}}kn(m+n)\mathrm{WhittakerM}(1/2\frac{m+n}{n},1/2\frac{m+2n}{n},\frac{x^{n}c}{n})+\mathit{\_}\mathit{F}\mathit{1}(y{x}^{-a},z{x}^{-b})cm(m+2n))(m+n))$$

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6.9.4.11 [1951] Problem 11

problem number 1951

Added Jan 16, 2020.

Problem Chapter 9.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$x{w}_x+az{w}_y+by{w}_z=c{x}^{n}w+k{x}^m$$

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^n*w[x,y,z]+ k*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{c{x}^n}{n}}(-\frac{k{x}^m{\left(\frac{c{x}^n}{n}\right)}^{-\frac{m}{n}}\text{Gamma}(\frac{m}{n},\frac{c{x}^n}{n})}{n}+c_1(iy\sinh (\sqrt{a}\sqrt{b}\log (x))-\frac{i\sqrt{a}z\cosh (\sqrt{a}\sqrt{b}\log (x))}{\sqrt{b}},y\cosh (\sqrt{a}\sqrt{b}\log (x))-\frac{\sqrt{a}z\sinh (\sqrt{a}\sqrt{b}\log (x))}{\sqrt{b}}))\}\right\}$$

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^n*w(x,y,z)+ k*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=({\int }^y\frac{k}{\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}}{\left(x{(\sqrt{ab}y+az)}^{-\frac{\sqrt{ab}}{ab}}{\left(\frac{ab\mathit{\_}\mathit{b}+\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}\sqrt{ab}}{\sqrt{ab}}\right)}^{\frac{1}{\sqrt{ab}}}\right)}^m{\mathrm{e}}^{-c\int \frac{1}{\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}}{\left(x{(\sqrt{ab}y+az)}^{-\frac{\sqrt{ab}}{ab}}{\left(\frac{ab\mathit{\_}\mathit{b}+\sqrt{a(a{z}^2+({\mathit{\_}\mathit{b}}^2-y^2)b)}\sqrt{ab}}{\sqrt{ab}}\right)}^{\frac{1}{\sqrt{ab}}}\right)}^n\mathrm{d}\mathit{\_}\mathit{b}}d\mathit{\_}\mathit{b}+\mathit{\_}\mathit{F}\mathit{1}(\frac{a{z}^2-b{y}^2}{a},x{(\sqrt{ab}y+az)}^{-\frac{\sqrt{ab}}{ab}})){\mathrm{e}}^{{\int }^y\frac{c}{\sqrt{a(a{z}^2+({\mathit{\_}\mathit{a}}^2-y^2)b)}}{\left(x{(\sqrt{ab}y+az)}^{-\frac{\sqrt{ab}}{ab}}{\left(\frac{ab\mathit{\_}\mathit{a}+\sqrt{a(a{z}^2+({\mathit{\_}\mathit{a}}^2-y^2)b)}\sqrt{ab}}{\sqrt{ab}}\right)}^{\frac{1}{\sqrt{ab}}}\right)}^{n}d\mathit{\_}\mathit{a}}$$

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6.9.4.12 [1952] Problem 12

problem number 1952

Added Jan 16, 2020.

Problem Chapter 9.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$bcx{w}_x+c(by+cz)w_y+b(by-cz)w_z=k{x}^{n}w+s{x}^m$$

Mathematica ✗

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

Failed

Maple ✓

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

$$w(x,y,z)=(-{\int }^y-\frac{s}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}c}{\left(x{\left((\frac{\sqrt{2}{b}^{2}y}{-b^2{y}^2+2bzcy+c^2{z}^2}+(\frac{by}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}+\frac{cz}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}})\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}})\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}\right)}^{-1/2\frac{b\sqrt{2}}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}}{\left((\frac{b^2\mathit{\_}\mathit{b}\sqrt{2}}{-b^2{y}^2+2bzcy+c^2{z}^2}+\sqrt{2\frac{b^2{\mathit{\_}\mathit{b}}^2}{-b^2{y}^2+2bzcy+c^2{z}^2}+1}\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}})\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}\right)}^{1/2\frac{b\sqrt{2}}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}}\right)}^m{\mathrm{e}}^{-\frac{k}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}c}\int {\left(x{\left((\frac{\sqrt{2}{b}^{2}y}{-b^2{y}^2+2bzcy+c^2{z}^2}+(\frac{by}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}+\frac{cz}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}})\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}})\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}\right)}^{-1/2\frac{b\sqrt{2}}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}}{\left((\frac{b^2\mathit{\_}\mathit{b}\sqrt{2}}{-b^2{y}^2+2bzcy+c^2{z}^2}+\sqrt{2\frac{b^2{\mathit{\_}\mathit{b}}^2}{-b^2{y}^2+2bzcy+c^2{z}^2}+1}\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}})\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}\right)}^{1/2\frac{b\sqrt{2}}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}}\right)}^n\frac{1}{\sqrt{2\frac{b^2{\mathit{\_}\mathit{b}}^2}{-b^2{y}^2+2bzcy+c^2{z}^2}+1}}\mathrm{d}\mathit{\_}\mathit{b}}\frac{1}{\sqrt{2\frac{b^2{\mathit{\_}\mathit{b}}^2}{-b^2{y}^2+2bzcy+c^2{z}^2}+1}}d\mathit{\_}\mathit{b}+\mathit{\_}\mathit{F}\mathit{1}(-\frac{1}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}},x{\left((\frac{\sqrt{2}{b}^{2}y}{-b^2{y}^2+2bzcy+c^2{z}^2}+(\frac{by}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}+\frac{cz}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}})\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}})\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}\right)}^{-1/2\frac{b\sqrt{2}}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}})){\mathrm{e}}^{-{\int }^y-\frac{k}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}c}{\left(x{\left((\frac{\sqrt{2}{b}^{2}y}{-b^2{y}^2+2bzcy+c^2{z}^2}+(\frac{by}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}+\frac{cz}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}})\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}})\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}\right)}^{-1/2\frac{b\sqrt{2}}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}}{\left((\frac{\mathit{\_}\mathit{a}{b}^2\sqrt{2}}{-b^2{y}^2+2bzcy+c^2{z}^2}+\sqrt{2\frac{{\mathit{\_}\mathit{a}}^2{b}^2}{-b^2{y}^2+2bzcy+c^2{z}^2}+1}\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}})\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}\right)}^{1/2\frac{b\sqrt{2}}{\sqrt{-b^2{y}^2+2bzcy+c^2{z}^2}}\frac{1}{\sqrt{\frac{b^2}{-b^2{y}^2+2bzcy+c^2{z}^2}}}}\right)}^n\frac{1}{\sqrt{2\frac{{\mathit{\_}\mathit{a}}^2{b}^2}{-b^2{y}^2+2bzcy+c^2{z}^2}+1}}d\mathit{\_}\mathit{a}}$$

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6.9.4.13 [1953] Problem 13

problem number 1953

Added Jan 16, 2020.

Problem Chapter 9.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$b_1{x}^{n_1}{w}_x+b_2{y}^{n_2}{w}_y+b_3{z}^{n_3}{w}_z=aw+c_1{x}^{k_1}+c_2{y}^{k_2}+c_3{x}^{k_3}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  b1*x^n1*D[w[x,y,z],x]+ b2*y^n2*D[w[x,y,z],y]+b3*z^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*x^k1+c2*y^k2+c3*x^k3; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{a{x}^{1-\text{n1}}}{\text{b1}-\text{b1}\text{n1}}}({\int }_1^x\frac{e^{\frac{aK[1{]}^{1-\text{n1}}}{\text{b1}(\text{n1}-1)}}K[1{]}^{-\text{n1}}(\text{c1}K[1{]}^{\text{k1}}+\text{c3}K[1{]}^{\text{k3}}+\text{c2}{\left({(\frac{\text{b2}(\text{n2}-1)x^{-\text{n1}}(x^{\text{n1}}K[1]-xK[1{]}^{\text{n1}})K[1{]}^{-\text{n1}}}{\text{b1}(\text{n1}-1)}+{\left(\frac{1}{y}\right)}^{\text{n2}-1})}^{\frac{1}{1-\text{n2}}}\right)}^{\text{k2}})}{\text{b1}}dK[1]+c_1(\frac{\text{b2}{x}^{1-\text{n1}}}{\text{b1}(\text{n1}-1)}-\frac{{\left(\frac{1}{y}\right)}^{\text{n2}-1}}{\text{n2}-1},\frac{\text{b3}{x}^{1-\text{n1}}}{\text{b1}(\text{n1}-1)}-\frac{{\left(\frac{1}{z}\right)}^{\text{n3}-1}}{\text{n3}-1}))\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   b__1*x^(n__1)*diff(w(x,y,z),x)+ b__2*y^(n__2)*diff(w(x,y,z),y)+ b__3*z^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*x^(k__1)+c__2*y^(k__2)+c__3*x^(k__3); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=({\int }^x\frac{1}{b_1}{\mathrm{e}}^{\frac{{\mathit{\_}\mathit{a}}^{-n_1+1}a}{(n_1-1)b_1}}({\mathit{\_}\mathit{a}}^{-n_1}{c}_2{\left({\left(\frac{-x^{-n_1+1}{b}_2(n_2-1)+y^{1-n_2}{b}_1(n_1-1)+{\mathit{\_}\mathit{a}}^{-n_1+1}{b}_2(n_2-1)}{(n_1-1)b_1}\right)}^{-{(n_2-1)}^{-1}}\right)}^{k_2}+{\mathit{\_}\mathit{a}}^{-n_1+k_1}{c}_1+{\mathit{\_}\mathit{a}}^{-n_1+k_3}{c}_3)d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}(\frac{-x^{-n_1+1}{b}_2(n_2-1)+y^{1-n_2}{b}_1(n_1-1)}{(n_1-1)b_1},\frac{-x^{-n_1+1}{b}_3(n_3-1)+z^{1-n_3}{b}_1(n_1-1)}{(n_1-1)b_1})){\mathrm{e}}^{-\frac{x^{-n_1+1}a}{(n_1-1)b_1}}$$

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6.9.4.14 [1954] Problem 14

problem number 1954

Added Jan 16, 2020.

Problem Chapter 9.2.4.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a_1{x}^{n_1}{w}_x+a_2{y}^{n_2}{w}_y+a_3{z}^{n_3}{w}_z=b{x}^{k}w+c{x}^m$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a1*x^n1*D[w[x,y,z],x]+ a2*y^n2*D[w[x,y,z],y]+a3*z^n3*D[w[x,y,z],z]==b*x^k*w[x,y,z]+ c*x^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\frac{b{x}^{k-\text{n1}+1}}{\text{a1}k-\text{a1}\text{n1}+\text{a1}}}(c_1(\frac{\text{a2}{x}^{1-\text{n1}}}{\text{a1}(\text{n1}-1)}-\frac{{\left(\frac{1}{y}\right)}^{\text{n2}-1}}{\text{n2}-1},\frac{\text{a3}{x}^{1-\text{n1}}}{\text{a1}(\text{n1}-1)}-\frac{{\left(\frac{1}{z}\right)}^{\text{n3}-1}}{\text{n3}-1})-\frac{c{x}^{m-\text{n1}+1}{\left(\frac{b{x}^{k-\text{n1}+1}}{\text{a1}k-\text{a1}\text{n1}+\text{a1}}\right)}^{\frac{-m+\text{n1}-1}{k-\text{n1}+1}}\text{Gamma}(\frac{m-\text{n1}+1}{k-\text{n1}+1},\frac{b{x}^{k-\text{n1}+1}}{\text{a1}k-\text{a1}\text{n1}+\text{a1}})}{\text{a1}(k-\text{n1}+1)})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   a__1*x^(n__1)*diff(w(x,y,z),x)+ a__2*y^(n__2)*diff(w(x,y,z),y)+ a__3*z^(n__3)*diff(w(x,y,z),z)=b*x^k*w(x,y,z)+ x*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=6\frac{1}{(-m+n_1-2)a_1(-k-m+2n_1-3)(-2k-m+3n_1-4)b}{\mathrm{e}}^{\frac{b{x}^{k-n_1+1}}{a_1(k-n_1+1)}}(a_1(-m+n_1-2)b(-k/2-m/2+n_1-3/2)(-2/3k-m/3+n_1-4/3)\mathit{\_}\mathit{F}\mathit{1}(\frac{-x^{-n_1+1}{a}_2(n_2-1)+y^{1-n_2}{a}_1(n_1-1)}{(n_1-1)a_1},\frac{-x^{-n_1+1}{a}_3(n_3-1)+z^{1-n_3}{a}_1(n_1-1)}{(n_1-1)a_1})-1/6{\left(\frac{b}{a_1(k-n_1+1)}\right)}^{\frac{m-n_1+2}{k-n_1+1}}{\left(\frac{b{x}^{k-n_1+1}}{a_1(k-n_1+1)}\right)}^{\frac{-k-m+2n_1-3}{2k-2n_1+2}}{\left(\frac{b}{a_1(k-n_1+1)}\right)}^{\frac{-m+n_1-2}{k-n_1+1}}(-4a_1{(-k/2-m/2+n_1-3/2)}^2\mathrm{WhittakerM}(\frac{k+m-2n_1+3}{2k-2n_1+2},\frac{2k+m-3n_1+4}{2k-2n_1+2},\frac{b{x}^{k-n_1+1}}{a_1(k-n_1+1)})+(b{x}^{k-n_1+1}-2a_1(-k/2-m/2+n_1-3/2))\mathrm{WhittakerM}(\frac{-k+1+m}{2k-2n_1+2},\frac{2k+m-3n_1+4}{2k-2n_1+2},\frac{b{x}^{k-n_1+1}}{a_1(k-n_1+1)})(-k+n_1-1)){\mathrm{e}}^{-1/2\frac{b{x}^{k-n_1+1}}{a_1(k-n_1+1)}}{x}^{-k+1+m}(-k+n_1-1))$$

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