Added Jan 18, 2020.
Problem Chapter 9.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c e^{\beta x} w +k e^{\lambda x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*D[w[x,y,z],y]+b*D[w[x,y,z],z]==c*Exp[beta*x]*w[x,y,z]+ k*Exp[lambda*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\beta x}}{\beta }} \left (\int _1^xe^{\lambda K[1]-\frac {c e^{\beta K[1]}}{\beta }} kdK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)=c*exp(beta*x)*w(x,y,z)+ k*exp(lambda*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!k{{\rm e}^{{\frac {\lambda \,x\beta -{{\rm e}^{\beta \,x}}c}{\beta }}}}\,{\rm d}x+{\it \_F1} \left ( -ax+y,-bx+z \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\beta \,x}}c}{\beta }}}}\]
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Added Jan 18, 2020.
Problem Chapter 9.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\beta x} w_y + b e^{\lambda x} w_z = c e^{\gamma x} w +s e^{\mu x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*Exp[beta*x]*D[w[x,y,z],y]+b*Exp[lambda*x]*D[w[x,y,z],z]==c*Exp[gamma*x]*w[x,y,z]+ s*Exp[mu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\gamma x}}{\gamma }} \left (\int _1^xe^{\mu K[1]-\frac {c e^{\gamma K[1]}}{\gamma }} sdK[1]+c_1\left (y-\frac {a e^{\beta x}}{\beta },z-\frac {b e^{\lambda x}}{\lambda }\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*exp(beta*x)*diff(w(x,y,z),y)+ b*exp(lambda*x)*diff(w(x,y,z),z)=c*exp(gamma*x)*w(x,y,z)+ s*exp(mu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!s{{\rm e}^{{\frac {\mu \,x\gamma -{{\rm e}^{\gamma \,x}}c}{\gamma }}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {-a{{\rm e}^{\beta \,x}}+y\beta }{\beta }},{\frac {z\lambda -b{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma \,x}}c}{\gamma }}}}\]
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Added Jan 18, 2020.
Problem Chapter 9.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + b e^{\beta x} w_y + c e^{\lambda y} w_z = a w +s e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ b*Exp[beta*x]*D[w[x,y,z],y]+c*Exp[lambda*y]*D[w[x,y,z],z]==a*w[x,y,z]+ s*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {e^{a x} \left ((\gamma -a) c_1\left (y-\frac {b e^{\beta x}}{\beta },z-\frac {c \text {ExpIntegralEi}\left (\frac {b \lambda e^{\beta x}}{\beta }\right ) e^{\lambda \left (y-\frac {b e^{\beta x}}{\beta }\right )}}{\beta }\right )+s e^{x (\gamma -a)}\right )}{a-\gamma }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ b*exp(beta*x)*diff(w(x,y,z),y)+ c*exp(lambda*y)*diff(w(x,y,z),z)=a*w(x,y,z)+ s*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {{{\rm e}^{ax}}}{a-\gamma } \left ( \left ( a-\gamma \right ) {\it \_F1} \left ( {\frac {-b{{\rm e}^{\beta \,x}}+y\beta }{\beta }},{\frac {1}{\beta } \left ( c{{\rm e}^{-{\frac { \left ( b{{\rm e}^{\beta \,x}}-y\beta \right ) \lambda }{\beta }}}}\Ei \left ( 1,-{\frac {\lambda \,b{{\rm e}^{\beta \,x}}}{\beta }} \right ) +z\beta \right ) } \right ) -s{{\rm e}^{-x \left ( a-\gamma \right ) }} \right ) }\]
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Added Jan 18, 2020.
Problem Chapter 9.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\beta x} w_y + b e^{\lambda z} w_z = c w +k e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*Exp[beta*x]*D[w[x,y,z],y]+b*Exp[lambda*z]*D[w[x,y,z],z]==c*w[x,y,z]+ k*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {e^{c x} \left ((\gamma -c) c_1\left (-\frac {b \lambda x+e^{-\lambda z}}{\lambda },y-\frac {a e^{\beta x}}{\beta }\right )+k e^{x (\gamma -c)}\right )}{c-\gamma }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*exp(beta*x)*diff(w(x,y,z),y)+ b*exp(lambda*z)*diff(w(x,y,z),z)=c*w(x,y,z)+ k*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( -{\frac {k{{\rm e}^{-x \left ( c-\gamma \right ) }}}{c-\gamma }}+{\it \_F1} \left ( {\frac {-a{{\rm e}^{\beta \,x}}+y\beta }{\beta }},{\frac {-b\lambda \,x-{{\rm e}^{-z\lambda }}}{\lambda \,b}} \right ) \right ) {{\rm e}^{cx}}\]
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Added Jan 18, 2020.
Problem Chapter 9.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\sigma x}+ a_2 e^{\lambda y} ) w_y + (b_1 e^{\mu y}+ b_2 e^{\beta z} ) w_z = c_1 w +c_2 e^{\nu x} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[sigma*x]+ a2*Exp[lambda*y] )*D[w[x,y,z],y]+(b1*Exp[mu*y]+ b2*Exp[beta*z])*D[w[x,y,z],z]==c1*w[x,y,z]+ c2*Exp[nu*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)+ (a__1*exp(sigma*x)+ a__2*exp(lambda*y) )*diff(w(x,y,z),y)+ (b__1*exp(mu*y)+ b__2*exp(beta*z))*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2*exp(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {{{\rm e}^{c_{1}\,x}}}{c_{1}-\nu } \left ( \left ( c_{1}-\nu \right ) {\it \_F1} \left ( {\frac {1}{\sigma \,\lambda } \left ( a_{2}\,\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\sigma \,x}}}{\sigma }} \right ) \lambda -{{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\sigma \,x}}-\sigma \,y \right ) }{\sigma }}}}\sigma \right ) },{\frac {1}{\beta } \left ( -\int ^{x}\!{{\rm e}^{b_{1}\,\beta \,\int \!{{\rm e}^{{\frac {\mu \,a_{1}\,{{\rm e}^{\sigma \,{\it \_f}}}}{\sigma }}}} \left ( {\frac {1}{\sigma } \left ( {{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\sigma \,x}}-\sigma \,y \right ) }{\sigma }}}}\sigma -a_{2}\,\lambda \, \left ( \Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\sigma \,x}}}{\sigma }} \right ) -\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\sigma \,{\it \_f}}}}{\sigma }} \right ) \right ) \right ) } \right ) ^{-{\frac {\mu }{\lambda }}}\,{\rm d}{\it \_f}}}{d{\it \_f}}b_{2}\,\beta -{{\rm e}^{\beta \, \left ( \int ^{x}\!{{\rm e}^{{\frac {\mu \,a_{1}\,{{\rm e}^{\sigma \,{\it \_b}}}}{\sigma }}}} \left ( {\frac {1}{\sigma } \left ( {{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\sigma \,x}}-\sigma \,y \right ) }{\sigma }}}}\sigma -a_{2}\,\lambda \, \left ( \Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\sigma \,x}}}{\sigma }} \right ) -\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\sigma \,{\it \_b}}}}{\sigma }} \right ) \right ) \right ) } \right ) ^{-{\frac {\mu }{\lambda }}}{d{\it \_b}}b_{1}-z \right ) }} \right ) } \right ) -c_{2}\,{{\rm e}^{x \left ( -c_{1}+\nu \right ) }} \right ) }\]
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Added Jan 18, 2020.
Problem Chapter 9.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ b_1 e^{\lambda _1 x} w_x + b_2 e^{\lambda _2 y} w_y + b_3 e^{\lambda _3 z} w_z = a w +c_1 e^{\beta _1 x}+c_2 e^{\beta _2 y}+c_3 e^{\beta _3 z} \]
Mathematica ✓
ClearAll["Global`*"]; pde = b1*Exp[lambda1*x]*D[w[x,y,z],x]+ b2*Exp[lambda2*y]*D[w[x,y,z],y]+b3*Exp[lambda3*z]*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Exp[beta1*x]+c2*Exp[beta2*y]+c3*Exp[beta3*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{-\frac {a e^{-\text {lambda1} x}}{\text {b1} \text {lambda1}}} \left (\int _1^x\frac {e^{\frac {a e^{-\text {lambda1} K[1]}}{\text {b1} \text {lambda1}}-\text {lambda1} K[1]} \left (\text {c2} \left (\frac {\text {b2} \left (-e^{-\text {lambda1} x}+e^{-\text {lambda1} K[1]}\right ) \text {lambda2}}{\text {b1} \text {lambda1}}+e^{-\text {lambda2} y}\right )^{-\frac {\text {beta2}}{\text {lambda2}}}+\text {c3} \left (\frac {\text {b3} \left (-e^{-\text {lambda1} x}+e^{-\text {lambda1} K[1]}\right ) \text {lambda3}}{\text {b1} \text {lambda1}}+e^{-\text {lambda3} z}\right )^{-\frac {\text {beta3}}{\text {lambda3}}}+\text {c1} e^{\text {beta1} K[1]}\right )}{\text {b1}}dK[1]+c_1\left (\frac {\text {b2} e^{-\text {lambda1} x}}{\text {b1} \text {lambda1}}-\frac {e^{-\text {lambda2} y}}{\text {lambda2}},\frac {\text {b3} e^{-\text {lambda1} x}}{\text {b1} \text {lambda1}}-\frac {e^{-\text {lambda3} z}}{\text {lambda3}}\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := b__1*exp(lambda__1*x)*diff(w(x,y,z),x)+ b__2*exp(lambda__2*y)*diff(w(x,y,z),y)+ b__3*exp(lambda__3*z)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*exp(beta__1*x)+ c__2*exp(beta__2*y)+ c__3*exp(beta__3*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int ^{x}\!{\frac {1}{b_{1}} \left ( c_{2}\, \left ( {\frac {b_{1}\,\lambda _{1}}{ \left ( b_{1}\,\lambda _{1}\,{{\rm e}^{\lambda _{1}\,x}}-{{\rm e}^{\lambda _{2}\,y}}b_{2}\,\lambda _{2} \right ) {{\rm e}^{-\lambda _{1}\,x-\lambda _{2}\,y}}+{{\rm e}^{-\lambda _{1}\,{\it \_a}}}b_{2}\,\lambda _{2}}} \right ) ^{{\frac {\beta _{2}}{\lambda _{2}}}}{{\rm e}^{{\frac {-{\lambda _{1}}^{2}{\it \_a}\,b_{1}+a{{\rm e}^{-\lambda _{1}\,{\it \_a}}}}{b_{1}\,\lambda _{1}}}}}+c_{3}\, \left ( {\frac {b_{1}\,\lambda _{1}}{ \left ( b_{1}\,\lambda _{1}\,{{\rm e}^{\lambda _{1}\,x}}-{{\rm e}^{\lambda _{3}\,z}}b_{3}\,\lambda _{3} \right ) {{\rm e}^{-\lambda _{1}\,x-\lambda _{3}\,z}}+{{\rm e}^{-\lambda _{1}\,{\it \_a}}}b_{3}\,\lambda _{3}}} \right ) ^{{\frac {\beta _{3}}{\lambda _{3}}}}{{\rm e}^{{\frac {-{\lambda _{1}}^{2}{\it \_a}\,b_{1}+a{{\rm e}^{-\lambda _{1}\,{\it \_a}}}}{b_{1}\,\lambda _{1}}}}}+c_{1}\,{{\rm e}^{{\frac {a{{\rm e}^{-\lambda _{1}\,{\it \_a}}}+{\it \_a}\,b_{1}\,\lambda _{1}\, \left ( \beta _{1}-\lambda _{1} \right ) }{b_{1}\,\lambda _{1}}}}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\lambda _{2}\,y}}b_{2}\,\lambda _{2}-b_{1}\,\lambda _{1}\,{{\rm e}^{\lambda _{1}\,x}} \right ) {{\rm e}^{-\lambda _{1}\,x-\lambda _{2}\,y}}}{b_{2}\,\lambda _{1}\,\lambda _{2}}},-{\frac { \left ( b_{1}\,\lambda _{1}\,{{\rm e}^{\lambda _{1}\,x}}-{{\rm e}^{\lambda _{3}\,z}}b_{3}\,\lambda _{3} \right ) {{\rm e}^{-\lambda _{1}\,x-\lambda _{3}\,z}}}{b_{3}\,\lambda _{1}\,\lambda _{3}}} \right ) \right ) {{\rm e}^{-{\frac {a{{\rm e}^{-\lambda _{1}\,x}}}{b_{1}\,\lambda _{1}}}}}\]
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Added Jan 18, 2020.
Problem Chapter 9.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 e^{\sigma _1 x+\beta _1 y} w_x + a_2 e^{\sigma _2 y+\beta _2 y} w_y + \left ( b_1 e^{\nu _1 x+\mu _1 y} + b_2 e^{\nu _2 x+\mu _2 y+ \lambda z} \right ) w_z = c_1 w +c_2 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Exp[sigma1*x+beta1*y]*D[w[x,y,z],x]+ a2*Exp[sigma2*y+beta2*y]*D[w[x,y,z],y]+( b1*Exp[nu1*x+mu1*y] + b2*Exp[nu2*x+mu2*y+lambda*z])*D[w[x,y,z],z]==c1*w[x,y,z]+ c2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
$Aborted
Maple ✓
restart; local gamma; pde := a__1*exp(sigma__1*x+beta__1*y)*diff(w(x,y,z),x)+ a__2*exp(sigma__2*y+beta__2*y)*diff(w(x,y,z),y)+ ( b__1*exp(nu__1*x+mu__1*y) + b__2*exp(nu__2*x+mu__2*y+lambda*z))*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int ^{x}\!{\frac {c_{2}}{a_{1}} \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}}{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}a_{1}\,\sigma _{1}+a_{2}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) \left ( {{\rm e}^{-\sigma _{1}\,x}}-{{\rm e}^{-\sigma _{1}\,{\it \_a}}} \right ) }{a_{1}\,\sigma _{1}}} \right ) ^{-{\frac {\beta _{1}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}{{\rm e}^{{\frac {1}{ \left ( \beta _{2}+\sigma _{2} \right ) a_{2}} \left ( c_{1}\, \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}}{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}a_{1}\,\sigma _{1}+a_{2}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) \left ( {{\rm e}^{-\sigma _{1}\,x}}-{{\rm e}^{-\sigma _{1}\,{\it \_a}}} \right ) }{a_{1}\,\sigma _{1}}} \right ) ^{{\frac {-\beta _{2}-\sigma _{2}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}-\sigma _{1}\,{\it \_a}\, \left ( \beta _{2}+\sigma _{2} \right ) a_{2} \right ) }}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}} \left ( a_{1}\,\sigma _{1}\,{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}+ \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) a_{2} \right ) }{\sigma _{1}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) a_{2}}},{\frac {1}{a_{1}\,\lambda } \left ( -\int ^{x}\! \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}}{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}a_{1}\,\sigma _{1}+a_{2}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) \left ( {{\rm e}^{-\sigma _{1}\,x}}-{{\rm e}^{-\sigma _{1}\,{\it \_g}}} \right ) }{a_{1}\,\sigma _{1}}} \right ) ^{{\frac {\mu _{2}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}{{\rm e}^{{\frac {1}{a_{1}} \left ( \lambda \,b_{1}\,\int \! \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}}{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}a_{1}\,\sigma _{1}+a_{2}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) \left ( {{\rm e}^{-\sigma _{1}\,x}}-{{\rm e}^{-\sigma _{1}\,{\it \_g}}} \right ) }{a_{1}\,\sigma _{1}}} \right ) ^{{\frac {-\beta _{1}+\mu _{1}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}{{\rm e}^{{\it \_g}\, \left ( \nu _{1}-\sigma _{1} \right ) }}\,{\rm d}{\it \_g}-{\it \_g}\,a_{1}\, \left ( \sigma _{1}-\nu _{2} \right ) \right ) }}} \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}}{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}a_{1}\,\sigma _{1}+a_{2}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) \left ( {{\rm e}^{-\sigma _{1}\,x}}-{{\rm e}^{-\sigma _{1}\,{\it \_b}}} \right ) }{a_{1}\,\sigma _{1}}} \right ) ^{-{\frac {\beta _{1}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}{d{\it \_g}}b_{2}\,\lambda -{{\rm e}^{{\frac {\lambda }{a_{1}} \left ( -za_{1}+\int ^{x}\! \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}}{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}a_{1}\,\sigma _{1}+a_{2}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) \left ( {{\rm e}^{-\sigma _{1}\,x}}-{{\rm e}^{-\sigma _{1}\,{\it \_b}}} \right ) }{a_{1}\,\sigma _{1}}} \right ) ^{{\frac {\mu _{1}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}{{\rm e}^{{\it \_b}\, \left ( \nu _{1}-\sigma _{1} \right ) }} \left ( {\frac {{{\rm e}^{-\sigma _{1}\,x}}{{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}a_{1}\,\sigma _{1}+a_{2}\, \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) \left ( {{\rm e}^{-\sigma _{1}\,x}}-{{\rm e}^{-\sigma _{1}\,{\it \_a}}} \right ) }{a_{1}\,\sigma _{1}}} \right ) ^{-{\frac {\beta _{1}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}{d{\it \_b}}b_{1} \right ) }}}a_{1} \right ) } \right ) \right ) {{\rm e}^{-{\frac {c_{1}}{ \left ( \beta _{2}+\sigma _{2} \right ) a_{2}} \left ( {{\rm e}^{y \left ( \beta _{1}-\beta _{2}-\sigma _{2} \right ) +\sigma _{1}\,x}}{{\rm e}^{-\sigma _{1}\,x}} \right ) ^{{\frac {-\beta _{2}-\sigma _{2}}{\beta _{1}-\beta _{2}-\sigma _{2}}}}}}}\]
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