6.7.5 3.1

6.7.5.1 [1615] Problem 1

problem number 1615

Added June 11, 2019.

Problem Chapter 7.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+a{e}^{\lambda x}{w}_y+b{e}^{\beta x}{w}_z=c{e}^{\gamma x}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*x]*D[w[x,y,z],z]== c*Exp[gamma*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to \frac{c{e}^{\gamma x}}{\gamma }+c_1(y-\frac{a{e}^{\lambda x}}{\lambda },z-\frac{b{e}^{\beta x}}{\beta })\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{c{\mathrm{e}}^{\gamma x}+\gamma \mathit{\_}\mathit{F}\mathit{1}(\frac{-a{\mathrm{e}}^{\lambda x}+\lambda y}{\lambda },\frac{-b{\mathrm{e}}^{\beta x}+\beta z}{\beta })}{\gamma }$$

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6.7.5.2 [1616] Problem 2

problem number 1616

Added June 11, 2019.

Problem Chapter 7.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+a{e}^{\lambda x}{w}_y+b{e}^{\beta y}{w}_z=c{e}^{\gamma y}+s{e}^{\mu z}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*y]*D[w[x,y,z],z]== c*Exp[gamma*y]+s*Exp[mu*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to {\int }_1^x(e^{\gamma (\frac{a(-e^{\lambda x}+e^{\lambda K[1]})}{\lambda }+y)}c+\exp \left(\frac{\mu (\lambda z-b{e}^{\beta (y-\frac{a{e}^{\lambda x}}{\lambda })}\text{Ei}\left(\frac{a\beta e^{\lambda x}}{\lambda }\right)+b{e}^{\beta (y-\frac{a{e}^{\lambda x}}{\lambda })}\text{Ei}\left(\frac{a\beta e^{\lambda K[1]}}{\lambda }\right))}{\lambda }\right)s)dK[1]+c_1(y-\frac{a{e}^{\lambda x}}{\lambda },z-\frac{b\text{Ei}\left(\frac{a\beta e^{\lambda x}}{\lambda }\right)e^{\beta (y-\frac{a{e}^{\lambda x}}{\lambda })}}{\lambda })\}\right\}$$

Maple ✓

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

$$w(x,y,z)={\int }_{}^x(c{\mathrm{e}}^{-\frac{(-a{\mathrm{e}}^{\mathit{\_}\mathit{a}\lambda }+a{\mathrm{e}}^{\lambda x}-\lambda y)\gamma }{\lambda }}+s{\mathrm{e}}^{\frac{((-\mathrm{expIntegral}(1,-\frac{a\beta {\mathrm{e}}^{\mathit{\_}\mathit{a}\lambda }}{\lambda })+\mathrm{expIntegral}(1,-\frac{a\beta {\mathrm{e}}^{\lambda x}}{\lambda }))b{\mathrm{e}}^{\frac{(-a{\mathrm{e}}^{\lambda x}+\lambda y)\beta }{\lambda }}+\lambda z)\mu }{\lambda }})d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}(\frac{-a{\mathrm{e}}^{\lambda x}+\lambda y}{\lambda },\frac{b\mathrm{expIntegral}(1,-\frac{a\beta {\mathrm{e}}^{\lambda x}}{\lambda }){\mathrm{e}}^{\frac{(-a{\mathrm{e}}^{\lambda x}+\lambda y)\beta }{\lambda }}+\lambda z}{\lambda })$$

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6.7.5.3 [1617] Problem 3

problem number 1617

Added June 11, 2019.

Problem Chapter 7.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+a{e}^{\lambda y}{w}_y+b{e}^{\beta y}{w}_z=c{e}^{\gamma x}+s{e}^{\mu z}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*y]*D[w[x, y,z], y] +b*Exp[beta*y]*D[w[x,y,z],z]== c*Exp[gamma*x]+s*Exp[mu*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to {\int }_1^x(e^{\gamma K[1]}c+\exp (-\frac{\mu (b\lambda (x-K[1]){(a\lambda (x-K[1])+e^{-\lambda y})}^{-\frac{\beta }{\lambda }}+(\beta -\lambda )z+\frac{b{e}^{-\lambda y}({(a\lambda (x-K[1])+e^{-\lambda y})}^{-\frac{\beta }{\lambda }}-{\left(e^{-\lambda y}\right)}^{-\frac{\beta }{\lambda }})}{a})}{\lambda -\beta })s)dK[1]+c_1(-\frac{a\lambda x+e^{-\lambda y}}{\lambda },\frac{b{\left(e^{-\lambda y}\right)}^{1-\frac{\beta }{\lambda }}}{a(\lambda -\beta )}+z)\}\right\}$$

Maple ✓

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

$$w(x,y,z)={\int }_{}^x(c{\mathrm{e}}^{\mathit{\_}\mathit{a}\gamma }+s{\mathrm{e}}^{\frac{(-b{\left({\mathrm{e}}^{\lambda y}\right)}^{\frac{\beta }{\lambda }}{\mathrm{e}}^{-\lambda y}+(\beta -\lambda )az+((-\mathit{\_}\mathit{a}+x)a\lambda +{\mathrm{e}}^{-\lambda y})b{\left(\frac{1}{(-\mathit{\_}\mathit{a}+x)a\lambda +{\mathrm{e}}^{-\lambda y}}\right)}^{\frac{\beta }{\lambda }})\mu }{(\beta -\lambda )a}})d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}(\frac{-a\lambda x-{\mathrm{e}}^{-\lambda y}}{a\lambda },\frac{-b{\left({\mathrm{e}}^{\lambda y}\right)}^{\frac{\beta }{\lambda }}{\mathrm{e}}^{-\lambda y}+(\beta -\lambda )az}{(\beta -\lambda )a})$$

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6.7.5.4 [1618] Problem 4

problem number 1618

Added June 11, 2019.

Problem Chapter 7.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+(A_1{e}^{{\alpha }_{1}x}+B_1{e}^{{\nu }_{1}x+\lambda y})w_y+(A_2{e}^{{\alpha }_{2}x}+B_2{e}^{{\nu }_{2}x+\beta y})w_z=k{e}^{\gamma z}$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (A1*Exp[alpha1*x] +B1*Exp[nu1*x+lambda*y] )*D[w[x, y,z], y] +(A2*Exp[alpha2*x] +B2*Exp[nu2*x+beta*y] )*D[w[x,y,z],z]== k*Exp[gamma*z]; 
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)+ (A1*exp(alpha1*x) +B1*exp(nu1*x+lambda*y) )*diff(w(x,y,z),y)+(A2*exp(alpha2*x) +B2*exp(nu2*x+beta*y) )*diff(w(x,y,z),z)= k*exp(gamma*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)={\int }_{}^{x}k{\mathrm{e}}^{(z+\int (\mathit{B}\mathit{2}{(-\mathit{B}\mathit{1}\lambda (\int {\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\lambda {\mathrm{e}}^{\mathit{\_}\mathit{f}\alpha 1}+\mathit{\_}\mathit{f}\alpha 1\nu 1}{\alpha 1}}d\mathit{\_}\mathit{f})+\mathit{B}\mathit{1}\lambda (\int {\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\lambda {\mathrm{e}}^{\alpha 1x}}{\alpha 1}+\nu 1x}dx)+{\mathrm{e}}^{\frac{(\mathit{A}\mathit{1}{\mathrm{e}}^{\alpha 1x}-\alpha 1y)\lambda }{\alpha 1}})}^{-\frac{\beta }{\lambda }}{\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\beta {\mathrm{e}}^{\mathit{\_}\mathit{f}\alpha 1}+\mathit{\_}\mathit{f}\alpha 1\nu 2}{\alpha 1}}+\mathit{A}\mathit{2}{\mathrm{e}}^{\mathit{\_}\mathit{f}\alpha 2})d\mathit{\_}\mathit{f}-\left({\int }_{}^x(\mathit{B}\mathit{2}{(-\mathit{B}\mathit{1}\lambda (\int {\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\lambda {\mathrm{e}}^{\mathit{\_}\mathit{b}\alpha 1}+\mathit{\_}\mathit{b}\alpha 1\nu 1}{\alpha 1}}d\mathit{\_}\mathit{b})+\mathit{B}\mathit{1}\lambda (\int {\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\lambda {\mathrm{e}}^{\alpha 1x}}{\alpha 1}+\nu 1x}dx)+{\mathrm{e}}^{\frac{(\mathit{A}\mathit{1}{\mathrm{e}}^{\alpha 1x}-\alpha 1y)\lambda }{\alpha 1}})}^{-\frac{\beta }{\lambda }}{\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\beta {\mathrm{e}}^{\mathit{\_}\mathit{b}\alpha 1}+\mathit{\_}\mathit{b}\alpha 1\nu 2}{\alpha 1}}+\mathit{A}\mathit{2}{\mathrm{e}}^{\mathit{\_}\mathit{b}\alpha 2})d\mathit{\_}\mathit{b}\right))\gamma }d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{F}\mathit{1}(\frac{-\mathit{B}\mathit{1}\lambda (\int {\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\lambda {\mathrm{e}}^{\alpha 1x}}{\alpha 1}+\nu 1x}dx)-{\mathrm{e}}^{\frac{(\mathit{A}\mathit{1}{\mathrm{e}}^{\alpha 1x}-\alpha 1y)\lambda }{\alpha 1}}}{\lambda },z-\left({\int }_{}^x(\mathit{B}\mathit{2}{(-\mathit{B}\mathit{1}\lambda (\int {\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\lambda {\mathrm{e}}^{\mathit{\_}\mathit{b}\alpha 1}+\mathit{\_}\mathit{b}\alpha 1\nu 1}{\alpha 1}}d\mathit{\_}\mathit{b})+\mathit{B}\mathit{1}\lambda (\int {\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\lambda {\mathrm{e}}^{\alpha 1x}}{\alpha 1}+\nu 1x}dx)+{\mathrm{e}}^{\frac{(\mathit{A}\mathit{1}{\mathrm{e}}^{\alpha 1x}-\alpha 1y)\lambda }{\alpha 1}})}^{-\frac{\beta }{\lambda }}{\mathrm{e}}^{\frac{\mathit{A}\mathit{1}\beta {\mathrm{e}}^{\mathit{\_}\mathit{b}\alpha 1}+\mathit{\_}\mathit{b}\alpha 1\nu 2}{\alpha 1}}+\mathit{A}\mathit{2}{\mathrm{e}}^{\mathit{\_}\mathit{b}\alpha 2})d\mathit{\_}\mathit{b}\right))$$

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6.7.5.5 [1619] Problem 5

problem number 1619

Added June 11, 2019.

Problem Chapter 7.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{e}^{\alpha x}{w}_x+b{e}^{\beta y}{w}_y+c{e}^{\gamma z}{w}_z=k{e}^{\lambda x}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[alpha*x]*D[w[x, y,z], x] + b*Exp[beta*y]*D[w[x, y,z], y] +c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to -\frac{k{e}^{x(\lambda -\alpha )}}{a(\alpha -\lambda )}+c_1(\frac{b{e}^{-\alpha x}}{a\alpha }-\frac{e^{-\beta y}}{\beta },\frac{c{e}^{-\alpha x}}{a\alpha }-\frac{e^{-\gamma z}}{\gamma })\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{(-\alpha +\lambda )a\mathit{\_}\mathit{F}\mathit{1}(-\frac{(a\alpha {\mathrm{e}}^{\alpha x}-b\beta {\mathrm{e}}^{\beta y}){\mathrm{e}}^{-\alpha x-\beta y}}{\alpha b\beta },-\frac{(a\alpha {\mathrm{e}}^{\alpha x}-c\gamma {\mathrm{e}}^{\gamma z}){\mathrm{e}}^{-\alpha x-\gamma z}}{\alpha c\gamma })+k{\mathrm{e}}^{-(\alpha -\lambda )x}}{(-\alpha +\lambda )a}$$

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6.7.5.6 [1620] Problem 6

problem number 1620

Added June 11, 2019.

Problem Chapter 7.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{e}^{\beta y}{w}_x+b{e}^{\alpha x}{w}_y+c{e}^{\gamma z}{w}_z=k{e}^{\lambda x}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[beta*x]*D[w[x, y,z], x] + b*Exp[alpha*x]*D[w[x, y,z], y] +c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to -\frac{k{e}^{x(\lambda -\beta )}}{a(\beta -\lambda )}+c_1(\frac{c{e}^{-\beta x}}{a\beta }-\frac{e^{-\gamma z}}{\gamma },y-\frac{b{e}^{\alpha x-\beta x}}{a\alpha -a\beta })\}\right\}$$

Maple ✓

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

$$w(x,y,z)={\int }_{}^x\frac{\alpha k{\mathrm{e}}^{\mathit{\_}\mathit{a}\lambda }}{a\alpha {\mathrm{e}}^{\beta y}-(-{\mathrm{e}}^{\mathit{\_}\mathit{a}\alpha }+{\mathrm{e}}^{\alpha x})b\beta }d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}(\frac{a\alpha {\mathrm{e}}^{\beta y}-b\beta {\mathrm{e}}^{\alpha x}}{\alpha b\beta },-\frac{(\alpha c\gamma x-c\gamma \ln \left(\frac{a\alpha {\mathrm{e}}^{\beta y}}{b\beta }\right)+(a\alpha {\mathrm{e}}^{\beta y}-b\beta {\mathrm{e}}^{\alpha x}){\mathrm{e}}^{-\gamma z})b\beta }{(a\alpha {\mathrm{e}}^{\beta y}-b\beta {\mathrm{e}}^{\alpha x})\alpha c\gamma })$$

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6.7.5.7 [1621] Problem 7

problem number 1621

Added June 11, 2019.

Problem Chapter 7.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$(a_1+a_2{e}^{\alpha x})w_x+(b_1+b_2{e}^{\beta y})w_y+(c_1+c_2{e}^{\gamma z})w_z=k_1+k_2{e}^{\alpha x}$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + (b1+b2*Exp[beta*y])*D[w[x, y,z], y] +(c1+c2*Exp[gamma*z])*D[w[x,y,z],z]== k1+k2*Exp[alpha*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to \frac{(\text{a1}\text{k2}-\text{a2}\text{k1})\log (\text{a1}+\text{a2}{e}^{\alpha x})+\text{a2}\alpha \text{k1}x}{\text{a1}\text{a2}\alpha }+c_1(\frac{\log \left(\frac{e^{\beta y}{(\text{a1}+\text{a2}{e}^{\alpha x})}^{\frac{\text{b1}\beta }{\text{a1}\alpha }}}{\text{b1}+\text{b2}{e}^{\beta y}}\right)}{\text{b1}\beta }-\frac{x}{\text{a1}},\frac{\log \left(\frac{e^{\gamma z}{(\text{a1}+\text{a2}{e}^{\alpha x})}^{\frac{\text{c1}\gamma }{\text{a1}\alpha }}}{\text{c1}+\text{c2}{e}^{\gamma z}}\right)}{\text{c1}\gamma }-\frac{x}{\text{a1}})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde := (a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+ (b1+b2*exp(beta*y))*diff(w(x,y,z),y)+(c1+c2*exp(gamma*z))*diff(w(x,y,z),z)= k1+k2*exp(alpha*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\frac{\mathit{a}\mathit{1}\mathit{a}\mathit{2}\alpha \mathit{\_}\mathit{F}\mathit{1}(\frac{-\mathit{a}\mathit{1}\alpha \mathrm{RootOf}(\mathit{a}\mathit{1}\alpha \beta y-\mathit{a}\mathit{1}\alpha \ln \left(\frac{(-\mathit{b}\mathit{1}+{\mathrm{e}}^{\mathit{\_}\mathit{Z}}){(\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})}^{\frac{\mathit{b}\mathit{1}\beta }{\mathit{a}\mathit{1}\alpha }}}{\mathit{b}\mathit{2}}\right)+\mathit{b}\mathit{1}\beta \ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1}))-(-\mathit{b}\mathit{1}\ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})+(-\mathit{a}\mathit{1}y+\mathit{b}\mathit{1}x)\alpha )\beta }{\mathit{a}\mathit{1}\alpha \mathit{b}\mathit{1}\beta },\frac{-\mathit{a}\mathit{1}\alpha \mathrm{RootOf}(\mathit{a}\mathit{1}\alpha \gamma z-\mathit{a}\mathit{1}\alpha \ln \left(\frac{(-\mathit{c}\mathit{1}+{\mathrm{e}}^{\mathit{\_}\mathit{Z}}){(\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})}^{\frac{\mathit{c}\mathit{1}\gamma }{\mathit{a}\mathit{1}\alpha }}}{\mathit{c}\mathit{2}}\right)+\mathit{c}\mathit{1}\gamma \ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1}))-(-\mathit{c}\mathit{1}\ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})+(-\mathit{a}\mathit{1}z+\mathit{c}\mathit{1}x)\alpha )\gamma }{\mathit{a}\mathit{1}\alpha \mathit{c}\mathit{1}\gamma })+\mathit{a}\mathit{1}\mathit{k}\mathit{2}\ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})-\mathit{a}\mathit{2}\mathit{k}\mathit{1}\ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})+\mathit{a}\mathit{2}\mathit{k}\mathit{1}\ln \left({\mathrm{e}}^{\alpha x}\right)}{\mathit{a}\mathit{1}\mathit{a}\mathit{2}\alpha }$$

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6.7.5.8 [1622] Problem 8

problem number 1622

Added June 11, 2019.

Problem Chapter 7.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$e^{\beta y}(a_1+a_2{e}^{\alpha x})w_x+e^{\alpha x}(b_1+b_2{e}^{\beta y})w_y+c{e}^{\beta y+\gamma z}{w}_z=k_3{e}^{\beta y}(k_1+k_2{e}^{\alpha x})$$

Mathematica ✓

ClearAll["Global`*"]; 
pde = Exp[beta*y]*(a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + Exp[alpha*x]*(b1+b2*Exp[beta*y])*D[w[x, y,z], y] +c*Exp[beta*y+gamma*z]*D[w[x,y,z],z]== k3*Exp[beta*y]*(k1+k2*Exp[alpha*x]); 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to \frac{\text{k3}((\text{a1}\text{k2}-\text{a2}\text{k1})\log (\text{a1}+\text{a2}{e}^{\alpha x})+\text{a2}\alpha \text{k1}x)}{\text{a1}\text{a2}\alpha }+c_1(\frac{c\log (\text{a1}+\text{a2}{e}^{\alpha x})}{\text{a1}\alpha }-\frac{cx}{\text{a1}}-\frac{e^{-\gamma z}}{\gamma },\frac{\log (\text{b1}+\text{b2}{e}^{\beta y})}{\text{b2}\beta }-\frac{\log (\text{a1}+\text{a2}{e}^{\alpha x})}{\text{a2}\alpha })\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde := exp(beta*y)*(a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+  exp(alpha*x)*(b1+b2*exp(beta*y))*diff(w(x,y,z),y)+c*exp(beta*y+gamma*z)*diff(w(x,y,z),z)= k3*exp(beta*y)*(k1+k2*exp(alpha*x)); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\frac{(\mathit{k}\mathit{1}\mathit{k}\mathit{3}x+\mathit{a}\mathit{1}\mathit{\_}\mathit{F}\mathit{1}(\frac{\mathit{a}\mathit{2}\alpha \beta y+\mathit{a}\mathit{2}\alpha \mathrm{RootOf}(\mathit{a}\mathit{2}\alpha \beta y-\mathit{a}\mathit{2}\alpha \ln \left(\frac{\mathit{b}\mathit{1}{(\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})}^{-\frac{\mathit{b}\mathit{2}\beta }{\mathit{a}\mathit{2}\alpha }}}{-\mathit{b}\mathit{2}+{\mathrm{e}}^{\mathit{\_}\mathit{Z}}}\right)-\mathit{b}\mathit{2}\beta \ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1}))-\mathit{b}\mathit{2}\beta \ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})}{\mathit{a}\mathit{2}\alpha \mathit{b}\mathit{2}\beta },\frac{-\alpha c\gamma x-\mathit{a}\mathit{1}\alpha {\mathrm{e}}^{-\gamma z}+c\gamma \ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})}{\mathit{a}\mathit{1}\alpha c\gamma }))\mathit{a}\mathit{2}\alpha +(\mathit{a}\mathit{1}\mathit{k}\mathit{2}-\mathit{a}\mathit{2}\mathit{k}\mathit{1})\mathit{k}\mathit{3}\ln (\mathit{a}\mathit{2}{\mathrm{e}}^{\alpha x}+\mathit{a}\mathit{1})}{\mathit{a}\mathit{1}\mathit{a}\mathit{2}\alpha }$$

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