6.7.24 8.1

6.7.24.1 [1722] Problem 1

problem number 1722

Added June 27, 2019.

Problem Chapter 7.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + f(x) w_y + g(x) w_z = h_2(x) y+h_1(x) + h_0(x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + f[x]*D[w[x, y,z], y] + g[x]*D[w[x,y,z],z]== h2[x]*y+h1[x]*z+h0[x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\left (\text {h0}(K[3])+\text {h2}(K[3]) \left (y-\int _1^xf(K[1])dK[1]+\int _1^{K[3]}f(K[1])dK[1]\right )+\text {h1}(K[3]) \left (z-\int _1^xg(K[2])dK[2]+\int _1^{K[3]}g(K[2])dK[2]\right )\right )dK[3]+c_1\left (y-\int _1^xf(K[1])dK[1],z-\int _1^xg(K[2])dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ f(x)*diff(w(x,y,z),y)+ g(x)*diff(w(x,y,z),z)= h2(x)*y+h1(x)*z+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int ^{x}\!{\it h2} \left ( {\it \_f} \right ) \int \!f \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it h1} \left ( {\it \_f} \right ) \int \!g \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it h2} \left ( {\it \_f} \right ) \left ( -\int \!f \left ( x \right ) \,{\rm d}x+y \right ) +{\it h1} \left ( {\it \_f} \right ) \left ( -\int \!g \left ( x \right ) \,{\rm d}x+z \right ) +{\it h0} \left ( {\it \_f} \right ) {d{\it \_f}}+{\it \_F1} \left ( -\int \!f \left ( x \right ) \,{\rm d}x+y,-\int \!g \left ( x \right ) \,{\rm d}x+z \right ) \]

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6.7.24.2 [1723] Problem 2

problem number 1723

Added June 27, 2019.

Problem Chapter 7.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + f(x)(y+a) w_y + g(x) (z+b)w_z = h(x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \int _1^xh(K[5])dK[5]+c_1\left (y \exp \left (-\int _1^xf(K[1])dK[1]\right )-\int _1^xa \exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) f(K[2])dK[2],z \exp \left (-\int _1^xg(K[3])dK[3]\right )-\int _1^xb \exp \left (-\int _1^{K[4]}g(K[3])dK[3]\right ) g(K[4])dK[4]\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y,z \right ) =\int \!h \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( \left ( y+a \right ) {{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}, \left ( z+b \right ) {{\rm e}^{-\int \!g \left ( x \right ) \,{\rm d}x}} \right ) \]

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6.7.24.3 [1724] Problem 3

problem number 1724

Added June 27, 2019.

Problem Chapter 7.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+f(x)) w_y + (b z+g(x))w_z = h(x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \int _1^xh(K[3])dK[3]+c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} g(K[2])dK[2]\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y,z \right ) =\int \!h \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( -\int \!f \left ( x \right ) {{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}},-\int \!g \left ( x \right ) {{\rm e}^{-xb}}\,{\rm d}x+z{{\rm e}^{-xb}} \right ) \]

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6.7.24.4 [1725] Problem 4

problem number 1725

Added June 27, 2019.

Problem Chapter 7.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y+f_2(x)) w_y + (g_1(x) y+g_2(x))w_z = h_2(x) y + h_1(x) z +h_0(x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] + (g1[x]*y+g2[x])*D[w[x,y,z],z]== h2[x]*y+h1[x]*z+h0[x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \text {h0}(K[5])+\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \text {h2}(K[5]) \left (y-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^x\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^{K[5]}\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]\right )+\text {h1}(K[5]) \left (-y \int _1^x\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+\left (y-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^x\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^{K[5]}\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]\right ) \int _1^{K[5]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \left (z-\int _1^x\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]+\int _1^{K[5]}\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]\right )\right )\right )dK[5]+c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3],-\int _1^x\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]-y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^x\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+z\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+ (g1(x)*y+g2(x))*diff(w(x,y,z),z)= h2(x)*y+h1(x)*z+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int ^{x}\!{\it h2} \left ( {\it \_g} \right ) y{{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}-{\it h2} \left ( {\it \_g} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{\it h2} \left ( {\it \_g} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\,{\rm d}{\it \_g}+{\it h1} \left ( {\it \_g} \right ) z+{\it h1} \left ( {\it \_g} \right ) \int \!{{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}y{\it g1} \left ( {\it \_g} \right ) -{{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}{\it g1} \left ( {\it \_g} \right ) \int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}{\it g1} \left ( {\it \_g} \right ) \int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\,{\rm d}{\it \_g}+{\it g2} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}-{\it h1} \left ( {\it \_g} \right ) \int ^{x}\!{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}y{\it g1} \left ( {\it \_f} \right ) -{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it g1} \left ( {\it \_f} \right ) \int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it g1} \left ( {\it \_f} \right ) \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}+{\it g2} \left ( {\it \_f} \right ) {d{\it \_f}}+{\it h0} \left ( {\it \_g} \right ) {d{\it \_g}}+{\it \_F1} \left ( -\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}},-\int ^{x}\!{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}y{\it g1} \left ( {\it \_f} \right ) -{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it g1} \left ( {\it \_f} \right ) \int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it g1} \left ( {\it \_f} \right ) \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}+{\it g2} \left ( {\it \_f} \right ) {d{\it \_f}}+z \right ) \]

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6.7.24.5 [1726] Problem 5

problem number 1726

Added June 27, 2019.

Problem Chapter 7.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y+f_2(x)) w_y + (g_1(x) z+g_2(x))w_z = h_2(x) y + h_1(x) z +h_0(x) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*z+f2[x])*D[w[x, y,z], y] + (g1[x]*y+g2[x])*D[w[x,y,z],z]== h2[x]*y+h1[x]*z+h0[x]; 
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)+ (f1(x)*z+f2(x))*diff(w(x,y,z),y)+ (g1(x)*y+g2(x))*diff(w(x,y,z),z)= h2(x)*y+h1(x)*z+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.7.24.6 [1727] Problem 6

problem number 1727

Added June 27, 2019.

Problem Chapter 7.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (y^2-a^2+a \lambda \sinh (\lambda x)-a^2 \sinh ^2(\lambda x)) w_y + f(x) \sinh (\gamma z) w_z = g(x) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (y^2-a^2 + a *lambda*Sin[lambda*x]-a^2*Sinh[lambda*x]^2)*D[w[x, y,z], y] + f[x]*Sinh[gamma*z]*D[w[x,y,z],z]== g[x]; 
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)+ (y^2-a^2 + a *lambda*sin(lambda*x)-a^2*sinh(lambda*x)^2)*diff(w(x,y,z),y)+ f(x)*sinh(gamma*z)*diff(w(x,y,z),z)= g(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[{\it PDESolStruc} \left ( w \left ( x,y,z \right ) ={\it \_F1} \left ( x \right ) +{\it \_F2} \left ( y \right ) -2\,{\frac {{\it \_c}_{{3}}\arctanh \left ( {{\rm e}^{\gamma \,z}} \right ) }{\gamma }}+{\it \_C1},[ \left \{ -{\frac {{a}^{2}{\frac {\rm d}{{\rm d}y}}{\it \_F2} \left ( y \right ) }{2}}+ \left ( {\frac {\rm d}{{\rm d}y}}{\it \_F2} \left ( y \right ) \right ) {y}^{2}-g \left ( x \right ) +{\frac {\rm d}{{\rm d}x}}{\it \_F1} \left ( x \right ) -{\frac {{a}^{2} \left ( {\frac {\rm d}{{\rm d}y}}{\it \_F2} \left ( y \right ) \right ) \cosh \left ( 2\,x\lambda \right ) }{2}}+\sin \left ( x\lambda \right ) a\lambda \,{\frac {\rm d}{{\rm d}y}}{\it \_F2} \left ( y \right ) +f \left ( x \right ) {\it \_c}_{{3}}=0 \right \} ] \right ) \] Gives Warning: Incomplete separation

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6.7.24.7 [1728] Problem 7

problem number 1728

Added June 27, 2019.

Problem Chapter 7.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y + f_2(x) y^k) w_y + (g_1(x) z+g_2(x) z^m) w_z = h(x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] + (g1[x]*z+g2[x]*z^m)*D[w[x,y,z],z]== h[x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^xh(K[5])dK[5]+c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x)*z+g2(x)*z^m)*diff(w(x,y,z),z)= h(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int \!h \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}, \left ( m-1 \right ) \int \!{{\rm e}^{ \left ( m-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}}{\it g2} \left ( x \right ) \,{\rm d}x+{z}^{1-m}{{\rm e}^{ \left ( m-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]

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6.7.24.8 [1729] Problem 8

problem number 1729

Added June 27, 2019.

Problem Chapter 7.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y + f_2(x) y^k) w_y + (g_1(x) +g_2(x) e^{\lambda z}) w_z = h(x) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] + (g1[x]+g2[x]*Exp[lambda*z])*D[w[x,y,z],z]== h[x]; 
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)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x)+g2(x)*exp(lambda*z))*diff(w(x,y,z),z)= h(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int \!h \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}},{\frac {-\lambda \,\int \!{\it g2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{\lambda \, \left ( \int \!{\it g1} \left ( x \right ) \,{\rm d}x-z \right ) }}}{\lambda }} \right ) \]

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6.7.24.9 [1730] Problem 9

problem number 1730

Added June 27, 2019.

Problem Chapter 7.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x)+ f_2(x) e^{\lambda y}) w_y + (g_1(x) +g_2(x) e^{\beta z}) w_z = h(x) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] + (g1[x]+g2[x]*Exp[beta*z])*D[w[x,y,z],z]== h[x]; 
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)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g1(x)+g2(x)*exp(beta*z))*diff(w(x,y,z),z)= h(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int \!h \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( {\frac {-\lambda \,\int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}}{\lambda }},{\frac {-\beta \,\int \!{\it g2} \left ( x \right ) {{\rm e}^{\beta \,\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{\beta \, \left ( \int \!{\it g1} \left ( x \right ) \,{\rm d}x-z \right ) }}}{\beta }} \right ) \]

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