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'));
 

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

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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'));
 

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

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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'));
 

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

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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