Added June 27, 2019.
Problem Chapter 7.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + g(y) w_y + h(z) w_z = \Phi (x) + \Psi (x) + \chi (x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y,z], x] + g[y]*D[w[x, y,z], y] + h[z]*D[w[x,y,z],z]== phi[x]+psi[x]+chi[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := f(x)*diff(w(x,y,z),x)+ g(y)*diff(w(x,y,z),y)+ h(z)*diff(w(x,y,z),z)= phi(x)+psi(x)+chi(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!{\frac {\phi \left ( x \right ) +\psi \left ( x \right ) +\chi \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y,-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( h \left ( z \right ) \right ) ^{-1}\,{\rm d}z \right ) \]
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Added June 27, 2019.
Problem Chapter 7.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + z w_y + g(y) w_z = h_2(x)+h_1(y) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y,z], x] + z*D[w[x, y,z], y] + g[y]*D[w[x,y,z],z]== h2[x]+h1[y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := f(x)*diff(w(x,y,z),x)+ z*diff(w(x,y,z),y)+ g(y)*diff(w(x,y,z),z)= h2(x)+h1(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{y}\!{\frac {{\it h2} \left ( \RootOf \left ( \int \!{\frac {1}{\sqrt {2\,\int \!g \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}\,{\rm d}{\it \_g}-\int ^{{\it \_Z}}\! \left ( f \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-\int ^{y}\!{\frac {1}{\sqrt {2\,\int \!g \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}{d{\it \_b}}+\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x \right ) \right ) +{\it h1} \left ( {\it \_g} \right ) }{\sqrt {2\,\int \!g \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}{d{\it \_g}}+{\it \_F1} \left ( {z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y,-\int ^{y}\!{\frac {1}{\sqrt {2\,\int \!g \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}{d{\it \_b}}+\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x \right ) \]
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Added June 27, 2019.
Problem Chapter 7.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) w_x + f_2(x) g(y) w_y + f_3(x) h(z) w_z = f_4(x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f1[x]*D[w[x, y,z], x] + f2[x]*g[y]*D[w[x, y,z], y] + f3[x]*h[z]*D[w[x,y,z],z]== f4[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := f1(x)*diff(w(x,y,z),x)+ f2(x)*g(y)*diff(w(x,y,z),y)+ f3(x)*h(z)*diff(w(x,y,z),z)= f4(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!{\frac {{\it f4} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( -\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y,-\int \!{\frac {{\it f3} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \! \left ( h \left ( z \right ) \right ) ^{-1}\,{\rm d}z \right ) \]
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Added June 27, 2019.
Problem Chapter 7.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x)) w_y + (g_1(x) z+g_2(y)) w_z = h_1(x)+h_2(y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] + (g1[x]*z+g2[y])*D[w[x,y,z],z]== h1[x]+h2[y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}\left (\exp \left (\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[4]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )+\int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[5]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )\right )dK[5]\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+ (g1(x)*z+g2(y))*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\it h1} \left ( {\it \_f} \right ) +{\it h2} \left ( \left ( \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}+{\it \_F1} \left ( -\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}},-\int ^{x}\!{\it g2} \left ( \left ( \int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\,{\rm d}{\it \_g}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}{d{\it \_g}}+z{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]
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Added June 27, 2019.
Problem Chapter 7.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(y) z^m) w_z = h_1(x)+h_2(y) \]
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[y]*z^m)*D[w[x,y,z],z]== h1[x]+h2[y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[4]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )+\int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[5]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )\right )dK[5]\right \}\right \}\]
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(y)*z^m)*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\it h1} \left ( {\it \_f} \right ) +{\it h2} \left ( \left ( \left ( 1-k \right ) \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f} \left ( k-1 \right ) }}\,{\rm d}{\it \_f}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}, \left ( m-1 \right ) \int ^{x}\!{{\rm e}^{\int \!{\it g1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g} \left ( m-1 \right ) }}{\it g2} \left ( {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}} \left ( \left ( 1-k \right ) \int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g} \left ( k-1 \right ) }}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_g}}+{z}^{-m+1}{{\rm e}^{ \left ( m-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]
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Added June 27, 2019.
Problem Chapter 7.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(y) e^{\lambda z}) w_z = h_1(x)+h_2(y) \]
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[y]*Exp[lambda*z])*D[w[x,y,z],z]== h1[x]+h2[y]; 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)*z+g2(y)*exp(lambda*z))*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added June 27, 2019.
Problem Chapter 7.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g_1(x) z+g_2(y) z^k) w_z = h_1(x)+h_2(y) \]
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]*z+g2[y]*z^k)*D[w[x,y,z],z]== h1[x]+h2[y]; 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)*z+g2(y)*z^k)*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\it h1} \left ( {\it \_f} \right ) +{\it h2} \left ( {\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}{\lambda }} \right ) {d{\it \_f}}+{\it \_F1} \left ( {\frac {-\lambda \,\int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}}{\lambda }}, \left ( k-1 \right ) \int ^{x}\!{\it g2} \left ( {\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\,{\rm d}{\it \_g} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}{\lambda }} \right ) {{\rm e}^{\int \!{\it g1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g} \left ( k-1 \right ) }}{d{\it \_g}}+{z}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]
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Added June 27, 2019.
Problem Chapter 7.8.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g_1(x) +g_2(y) e^{\beta z}) w_z = h_1(x)+h_2(y) \]
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[y]*Exp[beta*z])*D[w[x,y,z],z]== h1[x]+h2[y]; 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(y)*exp(beta*z))*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\it h1} \left ( {\it \_f} \right ) +{\it h2} \left ( {\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}{\lambda }} \right ) {d{\it \_f}}+{\it \_F1} \left ( {\frac {-\lambda \,\int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}}{\lambda }},{\frac {1}{\beta } \left ( -\beta \,\int ^{x}\!{\it g2} \left ( {\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}\,{\rm d}{\it \_g} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}{\lambda }} \right ) {{\rm e}^{\beta \,\int \!{\it g1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}{d{\it \_g}}-{{\rm e}^{\beta \, \left ( \int \!{\it g1} \left ( x \right ) \,{\rm d}x-z \right ) }} \right ) } \right ) \]
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