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'));
\[w \left (x , y , z\right ) = \left (\int k \,{\mathrm e}^{\frac {\beta \lambda x -c \,{\mathrm e}^{\beta x}}{\beta }}d x +\mathit {\_F1} \left (-a x +y , -x b +z \right )\right ) {\mathrm e}^{\frac {c \,{\mathrm e}^{\beta x}}{\beta }}\]
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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'));
\[w \left (x , y , z\right ) = \left (\int s \,{\mathrm e}^{\frac {\gamma \mu x -c \,{\mathrm e}^{\gamma x}}{\gamma }}d x +\mathit {\_F1} \left (\frac {-a \,{\mathrm e}^{\beta x}+\beta y}{\beta }, \frac {-b \,{\mathrm e}^{\lambda x}+\lambda z}{\lambda }\right )\right ) {\mathrm e}^{\frac {c \,{\mathrm e}^{\gamma x}}{\gamma }}\]
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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]];
\[\left \{\left \{w(x,y,z)\to -\frac {e^{a x} \left (s e^{x (\gamma -a)}+(\gamma -a) c_1\left (y-\frac {b e^{\beta x}}{\beta },z-\frac {c \text {Ei}\left (\frac {b e^{\beta x} \lambda }{\beta }\right ) e^{\lambda \left (y-\frac {b e^{\beta x}}{\beta }\right )}}{\beta }\right )\right )}{a-\gamma }\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \frac {\left (-s \,{\mathrm e}^{-\left (a -\gamma \right ) x}+\left (a -\gamma \right ) \mathit {\_F1} \left (\frac {-b \,{\mathrm e}^{\beta x}+\beta y}{\beta }, \frac {c \Ei \left (1, -\frac {b \lambda \,{\mathrm e}^{\beta x}}{\beta }\right ) {\mathrm e}^{-\frac {\left (b \,{\mathrm e}^{\beta x}-\beta y \right ) \lambda }{\beta }}+\beta z}{\beta }\right )\right ) {\mathrm e}^{a x}}{a -\gamma }\]
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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]];
\[\left \{\left \{w(x,y,z)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (-\frac {b \lambda x+e^{-\lambda z}}{\lambda },y-\frac {a e^{\beta x}}{\beta }\right )\right )}{c-\gamma }\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \left (-\frac {k \,{\mathrm e}^{-\left (c -\gamma \right ) x}}{c -\gamma }+\mathit {\_F1} \left (\frac {-a \,{\mathrm e}^{\beta x}+\beta y}{\beta }, \frac {-b \lambda x -{\mathrm e}^{-\lambda z}}{b \lambda }\right )\right ) {\mathrm e}^{c x}\]
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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'));
\[w \left (x , y , z\right ) = \frac {\left (-c_{2} \,{\mathrm e}^{\left (-c_{1} +\nu \right ) x}+\left (c_{1} -\nu \right ) \mathit {\_F1} \left (\frac {a_{2} \lambda \Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\sigma x}}{\sigma }\right )-\sigma \,{\mathrm e}^{\frac {\left (a_{1} \,{\mathrm e}^{\sigma x}-\sigma y \right ) \lambda }{\sigma }}}{\lambda \sigma }, \frac {-b_{2} \beta \left (\int _{}^{x}{\mathrm e}^{b_{1} \beta \left (\int \left (\frac {-\left (-\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\mathit {\_b} \sigma }}{\sigma }\right )+\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\sigma x}}{\sigma }\right )\right ) a_{2} \lambda +\sigma \,{\mathrm e}^{\frac {\left (a_{1} \,{\mathrm e}^{\sigma x}-\sigma y \right ) \lambda }{\sigma }}}{\sigma }\right )^{-\frac {\mu }{\lambda }} {\mathrm e}^{\frac {a_{1} \mu \,{\mathrm e}^{\mathit {\_b} \sigma }}{\sigma }}d \mathit {\_b} \right )}d\mathit {\_b} \right )-{\mathrm e}^{\left (b_{1} \left (\int _{}^{x}\left (\frac {-\left (-\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\mathit {\_b} \sigma }}{\sigma }\right )+\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\sigma x}}{\sigma }\right )\right ) a_{2} \lambda +\sigma \,{\mathrm e}^{\frac {\left (a_{1} \,{\mathrm e}^{\sigma x}-\sigma y \right ) \lambda }{\sigma }}}{\sigma }\right )^{-\frac {\mu }{\lambda }} {\mathrm e}^{\frac {a_{1} \mu \,{\mathrm e}^{\mathit {\_b} \sigma }}{\sigma }}d\mathit {\_b} \right )-z \right ) \beta }}{\beta }\right )\right ) {\mathrm e}^{c_{1} x}}{c_{1} -\nu }\]
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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'));
\[w \left (x , y , z\right ) = \left (\int _{}^{x}\frac {c_{2} \left (\frac {b_{1} \lambda _{1}}{b_{2} \lambda _{2} \,{\mathrm e}^{-\mathit {\_a} \lambda _{1}}+\left (b_{1} \lambda _{1} \,{\mathrm e}^{x \lambda _{1}}-b_{2} \lambda _{2} \,{\mathrm e}^{y \lambda _{2}}\right ) {\mathrm e}^{-x \lambda _{1} -y \lambda _{2}}}\right )^{\frac {\beta _{2}}{\lambda _{2}}} {\mathrm e}^{\frac {-\mathit {\_a} b_{1} \lambda _{1}^{2}+a \,{\mathrm e}^{-\mathit {\_a} \lambda _{1}}}{b_{1} \lambda _{1}}}+c_{3} \left (\frac {b_{1} \lambda _{1}}{b_{3} \lambda _{3} \,{\mathrm e}^{-\mathit {\_a} \lambda _{1}}+\left (b_{1} \lambda _{1} \,{\mathrm e}^{x \lambda _{1}}-b_{3} \lambda _{3} \,{\mathrm e}^{z \lambda _{3}}\right ) {\mathrm e}^{-x \lambda _{1} -z \lambda _{3}}}\right )^{\frac {\beta _{3}}{\lambda _{3}}} {\mathrm e}^{\frac {-\mathit {\_a} b_{1} \lambda _{1}^{2}+a \,{\mathrm e}^{-\mathit {\_a} \lambda _{1}}}{b_{1} \lambda _{1}}}+c_{1} \,{\mathrm e}^{\frac {\left (\beta _{1} -\lambda _{1} \right ) \mathit {\_a} b_{1} \lambda _{1} +a \,{\mathrm e}^{-\mathit {\_a} \lambda _{1}}}{b_{1} \lambda _{1}}}}{b_{1}}d\mathit {\_a} +\mathit {\_F1} \left (-\frac {\left (b_{1} \lambda _{1} \,{\mathrm e}^{x \lambda _{1}}-b_{2} \lambda _{2} \,{\mathrm e}^{y \lambda _{2}}\right ) {\mathrm e}^{-x \lambda _{1} -y \lambda _{2}}}{b_{2} \lambda _{1} \lambda _{2}}, -\frac {\left (b_{1} \lambda _{1} \,{\mathrm e}^{x \lambda _{1}}-b_{3} \lambda _{3} \,{\mathrm e}^{z \lambda _{3}}\right ) {\mathrm e}^{-x \lambda _{1} -z \lambda _{3}}}{b_{3} \lambda _{1} \lambda _{3}}\right )\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{-x \lambda _{1}}}{b_{1} \lambda _{1}}}\]
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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} \left (\frac {a_{1} \sigma _{1} \,{\mathrm e}^{-x \sigma _{1}} {\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}+\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) \left (-{\mathrm e}^{-\mathit {\_a} \sigma _{1}}+{\mathrm e}^{-x \sigma _{1}}\right ) a_{2}}{a_{1} \sigma _{1}}\right )^{-\frac {\beta _{1}}{\beta _{1} -\beta _{2} -\sigma _{2}}} {\mathrm e}^{\frac {-\left (\beta _{2} +\sigma _{2} \right ) \mathit {\_a} a_{2} \sigma _{1} +c_{1} \left (\frac {a_{1} \sigma _{1} \,{\mathrm e}^{-x \sigma _{1}} {\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}+\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) \left (-{\mathrm e}^{-\mathit {\_a} \sigma _{1}}+{\mathrm e}^{-x \sigma _{1}}\right ) a_{2}}{a_{1} \sigma _{1}}\right )^{\frac {-\beta _{2} -\sigma _{2}}{\beta _{1} -\beta _{2} -\sigma _{2}}}}{\left (\beta _{2} +\sigma _{2} \right ) a_{2}}}}{a_{1}}d\mathit {\_a} +\mathit {\_F1} \left (\frac {\left (a_{1} \sigma _{1} \,{\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}+\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) a_{2} \right ) {\mathrm e}^{-x \sigma _{1}}}{\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) a_{2} \sigma _{1}}, \frac {-b_{2} \lambda \left (\int _{}^{x}\left (\frac {a_{1} \sigma _{1} \,{\mathrm e}^{-x \sigma _{1}} {\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}+\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) \left (-{\mathrm e}^{-\mathit {\_b} \sigma _{1}}+{\mathrm e}^{-x \sigma _{1}}\right ) a_{2}}{a_{1} \sigma _{1}}\right )^{\frac {-\beta _{1} +\mu _{2}}{\beta _{1} -\beta _{2} -\sigma _{2}}} {\mathrm e}^{\frac {b_{1} \lambda \left (\int \left (\frac {a_{1} \sigma _{1} \,{\mathrm e}^{-x \sigma _{1}} {\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}+\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) \left (-{\mathrm e}^{-\mathit {\_f} \sigma _{1}}+{\mathrm e}^{-x \sigma _{1}}\right ) a_{2}}{a_{1} \sigma _{1}}\right )^{\frac {-\beta _{1} +\mu _{1}}{\beta _{1} -\beta _{2} -\sigma _{2}}} {\mathrm e}^{\left (\nu _{1} -\sigma _{1} \right ) \mathit {\_f}}d \mathit {\_f} \right )-\left (-\nu _{2} +\sigma _{1} \right ) \mathit {\_f} a_{1}}{a_{1}}}d\mathit {\_f} \right )-a_{1} \,{\mathrm e}^{\frac {\left (-a_{1} z +b_{1} \left (\int _{}^{x}\left (\frac {a_{1} \sigma _{1} \,{\mathrm e}^{-x \sigma _{1}} {\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}+\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) \left (-{\mathrm e}^{-\mathit {\_a} \sigma _{1}}+{\mathrm e}^{-x \sigma _{1}}\right ) a_{2}}{a_{1} \sigma _{1}}\right )^{-\frac {\beta _{1}}{\beta _{1} -\beta _{2} -\sigma _{2}}} \left (\frac {a_{1} \sigma _{1} \,{\mathrm e}^{-x \sigma _{1}} {\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}+\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) \left (-{\mathrm e}^{-\mathit {\_f} \sigma _{1}}+{\mathrm e}^{-x \sigma _{1}}\right ) a_{2}}{a_{1} \sigma _{1}}\right )^{\frac {\mu _{1}}{\beta _{1} -\beta _{2} -\sigma _{2}}} {\mathrm e}^{\left (\nu _{1} -\sigma _{1} \right ) \mathit {\_f}}d\mathit {\_f} \right )\right ) \lambda }{a_{1}}}}{a_{1} \lambda }\right )\right ) {\mathrm e}^{-\frac {c_{1} \left ({\mathrm e}^{-x \sigma _{1}} {\mathrm e}^{x \sigma _{1} +\left (\beta _{1} -\beta _{2} -\sigma _{2} \right ) y}\right )^{\frac {-\beta _{2} -\sigma _{2}}{\beta _{1} -\beta _{2} -\sigma _{2}}}}{\left (\beta _{2} +\sigma _{2} \right ) a_{2}}}\]
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