Added Jan 19, 2020.
Problem Chapter 9.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c e^{\beta x} w + k x^n \]
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*x^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\beta x}}{\beta }} \left (\int _1^xe^{-\frac {c e^{\beta K[1]}}{\beta }} k K[1]^ndK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
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*x^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!{x}^{n}k{{\rm e}^{-{\frac {{{\rm e}^{\beta \,x}}c}{\beta }}}}\,{\rm d}x+{\it \_F1} \left ( -ax+y,-bx+z \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\beta \,x}}c}{\beta }}}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a x^n w_y + b e^{\lambda x} w_z = c e^{\gamma x} w + s x^k \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*x^n*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*x^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\gamma x}}{\gamma }} \left (\int _1^xe^{-\frac {c e^{\gamma K[1]}}{\gamma }} s K[1]^kdK[1]+c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},z-\frac {b e^{\lambda x}}{\lambda }\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*x^n*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*x^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!s{x}^{k}{{\rm e}^{-{\frac {{{\rm e}^{\gamma \,x}}c}{\gamma }}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {-xa{x}^{n}+y \left ( n+1 \right ) }{n+1}},{\frac {\lambda \,z-b{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma \,x}}c}{\gamma }}}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + b e^{\beta x} w_y + c y^n w_z = a w + s e^{\gamma x} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ b*Exp[beta*x]*D[w[x,y,z],y]+c*y^n*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]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ b*exp(beta*x)*diff(w(x,y,z),y)+ c*y^n*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'));
\[w \left ( x,y,z \right ) ={\frac {{{\rm e}^{ax}}}{a-\gamma } \left ( \left ( a-\gamma \right ) {\it \_F1} \left ( {\frac {-b{{\rm e}^{\beta \,x}}+\beta \,y}{\beta }},-\int ^{x}\!c \left ( {\frac {b{{\rm e}^{\beta \,{\it \_a}}}-b{{\rm e}^{\beta \,x}}+\beta \,y}{\beta }} \right ) ^{n}{d{\it \_a}}+z \right ) -s{{\rm e}^{-x \left ( a-\gamma \right ) }} \right ) }\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 y+a_2 x y^k) w_y + (b_1 x+b_2 e^{\beta y+\lambda z}) w_z = c_1 w + c_2 e^{\gamma x} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*y+a2*x*y^k)*D[w[x,y,z],y]+(b1*x+b2*Exp[beta*y+lambda*z])*D[w[x,y,z],z]==c1*w[x,y,z]+ c2*Exp[gamma*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*y+a__2*x*y^k)*diff(w(x,y,z),y)+ (b__1*x+b__2*exp(beta*y+lambda*z))*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {{{\rm e}^{c_{1}\,x}}}{c_{1}-\gamma } \left ( \left ( c_{1}-\gamma \right ) {\it \_F1} \left ( {\frac { \left ( \left ( -1+a_{1}\, \left ( k-1 \right ) x \right ) a_{2}\,{y}^{k}+y{a_{1}}^{2} \left ( k-1 \right ) \right ) {{\rm e}^{a_{1}\, \left ( k-1 \right ) x}}}{{y}^{k} \left ( k-1 \right ) {a_{1}}^{2}}},{\frac {1}{\lambda } \left ( -b_{2}\,\int ^{x}\!{{\rm e}^{\beta \, \left ( {\frac { \left ( \left ( -1+a_{1}\, \left ( k-1 \right ) x \right ) a_{2}\,{y}^{k}+y{a_{1}}^{2} \left ( k-1 \right ) \right ) {{\rm e}^{a_{1}\, \left ( x-{\it \_a} \right ) \left ( k-1 \right ) }}-a_{2}\, \left ( -1+a_{1}\, \left ( k-1 \right ) {\it \_a} \right ) {y}^{k}}{{y}^{k} \left ( k-1 \right ) {a_{1}}^{2}}} \right ) ^{- \left ( k-1 \right ) ^{-1}}+{\frac {b_{1}\,{{\it \_a}}^{2}\lambda }{2}}}}{d{\it \_a}}\lambda -{{\rm e}^{{\frac {\lambda \, \left ( b_{1}\,{x}^{2}-2\,z \right ) }{2}}}} \right ) } \right ) -c_{2}\,{{\rm e}^{-x \left ( c_{1}-\gamma \right ) }} \right ) }\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 x+a_2 e^{\lambda y}) w_y + (b_1 z+b_2 e^{\beta y} z^k) w_z = c_1 w + c_2 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*x+a2*Exp[lambda*y])*D[w[x,y,z],y]+(b1*z+b2*Exp[beta*y]*z^k)*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]];
\[\left \{\left \{w(x,y,z)\to -\frac {\text {c2}}{\text {c1}}+e^{\text {c1} x} c_1\left (-\frac {\frac {\sqrt {2 \pi } \text {a2} \sqrt {\lambda } \text {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )}{\text {a1}^{3/2}}+\frac {2 e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a1}}}{2 \text {a2} \lambda ^2},(k-1) \int _1^x\text {b2} \exp \left (\frac {1}{2} \text {a1} \beta K[1]^2+\text {b1} (k-1) K[1]-\frac {\beta \left (\log (\text {a1})+\log (\text {a2})+2 \log (\lambda )+\log \left (\frac {\sqrt {\lambda } \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )+\frac {2 \sqrt {\text {a1}} e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a2}}-\sqrt {\lambda } \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } K[1]}{\sqrt {2}}\right )}{2 \text {a1}^{3/2} \lambda ^2}\right )\right )}{\lambda }\right )dK[1]+z^{1-k} e^{\text {b1} (k-1) x}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*x+a__2*exp(lambda*y))*diff(w(x,y,z),y)+ (b__1*z+b__2*exp(beta*y)*z^k)*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'));
\[w \left ( x,y,z \right ) ={\frac {1}{c_{1}} \left ( {{\rm e}^{c_{1}\,x}}{\it \_F1} \left ( {\frac {1}{2\,a_{1}\,\lambda } \left ( \sqrt {\pi }\sqrt {2}\sqrt {-a_{1}\,\lambda }\erf \left ( {\frac {\sqrt {2}x}{2}\sqrt {-a_{1}\,\lambda }} \right ) a_{2}\,\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}+2\,a_{1} \right ) {\frac {1}{\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}}}},b_{2}\,{2}^{{\frac {\beta }{2\,\lambda }}}{\pi }^{-{\frac {\beta }{2\,\lambda }}} \left ( k-1 \right ) \int ^{x}\!{{\rm e}^{{\frac {\beta \,{{\it \_a}}^{2}a_{1}}{2}}+b_{1}\, \left ( k-1 \right ) {\it \_a}}}{\pi }^{{\frac {\beta }{\lambda }}} \left ( -4\,{{a_{1}}^{3}\lambda \left ( {\frac { \left ( \sqrt {\pi }\sqrt {2}\sqrt {-a_{1}\,\lambda }\erf \left ( 1/2\,\sqrt {2}\sqrt {-a_{1}\,\lambda }x \right ) a_{2}\,\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}+2\,a_{1} \right ) \sqrt {-2\,a_{1}\,\lambda \,\pi }}{\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}}}+2\,a_{2}\,\erf \left ( 1/2\,\sqrt {-2\,a_{1}\,\lambda }{\it \_a} \right ) \pi \,a_{1}\,\lambda \right ) ^{-2}} \right ) ^{{\frac {\beta }{2\,\lambda }}}{d{\it \_a}}+{z}^{-k+1}{{\rm e}^{b_{1}\, \left ( k-1 \right ) x}} \right ) c_{1}-c_{2} \right ) }\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\mu x}+a_2 e^{\lambda y}) w_y + (b_1 e^{\nu y}+b_2 e^{\beta z}) w_z = c_1 w + c_2 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[mu*x]+a2*Exp[lambda*y])*D[w[x,y,z],y]+(b1*Exp[nu*y]+b2*Exp[beta*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]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*exp(mu*x)+a__2*exp(lambda*y))*diff(w(x,y,z),y)+ (b__1*exp(nu*y)+b__2*exp(beta*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'));
\[w \left ( x,y,z \right ) ={\frac {1}{c_{1}} \left ( {{\rm e}^{c_{1}\,x}}{\it \_F1} \left ( {\frac {1}{\mu \,\lambda } \left ( a_{2}\,\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,x}}}{\mu }} \right ) \lambda -{{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\mu \,x}}-\mu \,y \right ) }{\mu }}}}\mu \right ) },{\frac {1}{\beta } \left ( -\int ^{x}\!{{\rm e}^{b_{1}\,\beta \,\int \!{{\rm e}^{{\frac {\nu \,a_{1}\,{{\rm e}^{{\it \_b}\,\mu }}}{\mu }}}} \left ( {\frac {1}{\mu } \left ( {{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\mu \,x}}-\mu \,y \right ) }{\mu }}}}\mu -\lambda \,a_{2}\, \left ( \Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,x}}}{\mu }} \right ) -\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{{\it \_b}\,\mu }}}{\mu }} \right ) \right ) \right ) } \right ) ^{-{\frac {\nu }{\lambda }}}\,{\rm d}{\it \_b}}}{d{\it \_b}}b_{2}\,\beta -{{\rm e}^{\beta \, \left ( \int ^{x}\!{{\rm e}^{{\frac {\nu \,a_{1}\,{{\rm e}^{{\it \_b}\,\mu }}}{\mu }}}} \left ( {\frac {1}{\mu } \left ( {{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\mu \,x}}-\mu \,y \right ) }{\mu }}}}\mu -\lambda \,a_{2}\, \left ( \Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,x}}}{\mu }} \right ) -\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{{\it \_b}\,\mu }}}{\mu }} \right ) \right ) \right ) } \right ) ^{-{\frac {\nu }{\lambda }}}{d{\it \_b}}b_{1}-z \right ) }} \right ) } \right ) c_{1}-c_{2} \right ) }\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\lambda _1 x} y+a_2 e^{\lambda _2 x}) w_y + (b_1 e^{\beta _1 x}z+b_2 e^{\beta _2 x}) w_z = c_1 e^{\gamma _1 x} w + c_2 e^{\gamma _2 x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[lambda1*x]*y+a2*Exp[lambda2*x])*D[w[x,y,z],y]+(b1*Exp[beta1*x]*z+b2*Exp[beta2*x])*D[w[x,y,z],z]==c1*Exp[gamma1*x]*w[x,y,z]+ c2*Exp[gamma2*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {c1} e^{\text {gamma1} x}}{\text {gamma1}}} \left (\int _1^x\text {c2} e^{\text {gamma2} K[3]-\frac {\text {c1} e^{\text {gamma1} K[3]}}{\text {gamma1}}}dK[3]+c_1\left (y e^{-\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}}-\int _1^x\text {a2} e^{\text {lambda2} K[1]-\frac {\text {a1} e^{\text {lambda1} K[1]}}{\text {lambda1}}}dK[1],z e^{-\frac {\text {b1} e^{\text {beta1} x}}{\text {beta1}}}-\int _1^x\text {b2} e^{\text {beta2} K[2]-\frac {\text {b1} e^{\text {beta1} K[2]}}{\text {beta1}}}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*exp(lambda__1*x)*y+a__2*exp(lambda__2*x))*diff(w(x,y,z),y)+ (b__1*exp(beta__1*x)+b__2*exp(beta__2*x))*diff(w(x,y,z),z)=c__1*exp(gamma__1*x)*w(x,y,z)+ c__2*exp(gamma__2*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!c_{2}\,{{\rm e}^{{\frac {\gamma _{2}\,x\gamma _{1}-{{\rm e}^{\gamma _{1}\,x}}c_{1}}{\gamma _{1}}}}}\,{\rm d}x+{\it \_F1} \left ( -a_{2}\,\int \!{{\rm e}^{{\frac {\lambda _{2}\,x\lambda _{1}-{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{-{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}},{\frac {-b_{1}\,{{\rm e}^{\beta _{1}\,x}}\beta _{2}+\beta _{1}\, \left ( z\beta _{2}-b_{2}\,{{\rm e}^{\beta _{2}\,x}} \right ) }{\beta _{1}\,\beta _{2}}} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma _{1}\,x}}c_{1}}{\gamma _{1}}}}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\lambda _1 x} y+a_2 e^{\lambda _2 x} y^k) w_y + (b_1 e^{\beta _1 x}z+b_2 e^{\beta _2 x} z^m) w_z = c_1 e^{\gamma _1 x} w + c_2 e^{\gamma _2 y} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[lambda1*x]*y+a2*Exp[lambda2*x]*y^k)*D[w[x,y,z],y]+(b1*Exp[beta1*x]*z+b2*Exp[beta2*x]*z^m)*D[w[x,y,z],z]==c1*Exp[gamma1*x]*w[x,y,z]+ c2*Exp[gamma2*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {c1} e^{\text {gamma1} x}}{\text {gamma1}}} \left (\int _1^x\text {c2} \exp \left (\text {gamma2} \left (e^{-\frac {\text {a1} \left (e^{\text {lambda1} K[3]} (k-1)+e^{\text {lambda1} x}\right )}{\text {lambda1}}} y^{-k} \left (e^{\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}} (k-1) \int _1^x\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1] y^k-e^{\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}} (k-1) \int _1^{K[3]}\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1] y^k+e^{\frac {\text {a1} e^{\text {lambda1} x} k}{\text {lambda1}}} y\right )\right ){}^{\frac {1}{1-k}}-\frac {\text {c1} e^{\text {gamma1} K[3]}}{\text {gamma1}}\right )dK[3]+c_1\left ((k-1) \int _1^x\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {b1} e^{\text {beta1} K[2]} (m-1)}{\text {beta1}}+\text {beta2} K[2]}dK[2]+z^{1-m} e^{\frac {\text {b1} (m-1) e^{\text {beta1} x}}{\text {beta1}}}\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*exp(lambda__1*x)*y+a__2*exp(lambda__2*x)*y^k)*diff(w(x,y,z),y)+ (b__1*exp(beta__1*x)*z+b__2*exp(beta__2*x)*z^m)*diff(w(x,y,z),z)=c__1*exp(gamma__1*x)*w(x,y,z)+ c__2*exp(gamma__2*y); 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}\!c_{2}\,{{\rm e}^{{\frac {1}{\gamma _{1}} \left ( \gamma _{2}\, \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_b}}}+\lambda _{2}\,{\it \_b}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_b}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_b}}}a_{1}}{\lambda _{1}}}}}\gamma _{1}-{{\rm e}^{\gamma _{1}\,{\it \_b}}}c_{1} \right ) }}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}},{\frac {1}{{z}^{m}} \left ( {{\rm e}^{{\frac {b_{1}\,{{\rm e}^{\beta _{1}\,x}}}{\beta _{1}}}}}b_{2}\,{z}^{m} \left ( -1+m \right ) \int \!{{\rm e}^{{\frac {b_{1}\,{{\rm e}^{\beta _{1}\,x}} \left ( -1+m \right ) +\beta _{2}\,x\beta _{1}}{\beta _{1}}}}}\,{\rm d}x+z{{\rm e}^{{\frac {b_{1}\,{{\rm e}^{\beta _{1}\,x}}m}{\beta _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {b_{1}\,{{\rm e}^{\beta _{1}\,x}}}{\beta _{1}}}}} \right ) ^{-1}} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma _{1}\,x}}c_{1}}{\gamma _{1}}}}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\lambda _1 x} y+a_2 e^{\lambda _2 x} y^k) w_y + (b_1 e^{\beta _1 y}z+b_2 e^{\beta _2 y} z^m) w_z = c_1 e^{\gamma _1 x} w + c_2 e^{\gamma _2 z} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[lambda1*x]*y+a2*Exp[lambda2*x]*y^k)*D[w[x,y,z],y]+(b1*Exp[beta1*y]*z+b2*Exp[beta2*y]*z^m)*D[w[x,y,z],z]==c1*Exp[gamma1*x]*w[x,y,z]+ c2*Exp[gamma2*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
$Aborted
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*exp(lambda__1*x)*y+a__2*exp(lambda__2*x)*y^k)*diff(w(x,y,z),y)+ (b__1*exp(beta__1*y)*z+b__2*exp(beta__2*y)*z^m)*diff(w(x,y,z),z)=c__1*exp(gamma__1*x)*w(x,y,z)+ c__2*exp(gamma__2*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}\!c_{2}\,{{\rm e}^{{\frac {1}{\gamma _{1}} \left ( \gamma _{2}\, \left ( {z}^{1-m}{{\rm e}^{b_{1}\,\int ^{x}\!{{\rm e}^{\beta _{1}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_b}}}a_{1}}{\lambda _{1}}}}} \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_b}}}+\lambda _{2}\,{\it \_b}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_b}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}}}{d{\it \_b}} \left ( -1+m \right ) }}+ \left ( -1+m \right ) \left ( \int ^{x}\!{{\rm e}^{b_{1}\, \left ( -1+m \right ) \int \!{{\rm e}^{\beta _{1}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_b}}}a_{1}}{\lambda _{1}}}}} \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_b}}}+\lambda _{2}\,{\it \_b}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_b}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}}}\,{\rm d}{\it \_b}+ \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_f}}}+\lambda _{2}\,{\it \_f}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_f}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}\beta _{2}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_b}}}a_{1}}{\lambda _{1}}}}}}}{d{\it \_b}}-\int \!{{\rm e}^{b_{1}\, \left ( -1+m \right ) \int \!{{\rm e}^{\beta _{1}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_g}}}a_{1}}{\lambda _{1}}}}} \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_g}}}+\lambda _{2}\,{\it \_g}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_g}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}}}\,{\rm d}{\it \_g}+\beta _{2}\, \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_g}}}+\lambda _{2}\,{\it \_g}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_g}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_g}}}a_{1}}{\lambda _{1}}}}}}}\,{\rm d}{\it \_g} \right ) b_{2} \right ) ^{- \left ( -1+m \right ) ^{-1}}{{\rm e}^{b_{1}\,\int \!{{\rm e}^{\beta _{1}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_g}}}a_{1}}{\lambda _{1}}}}} \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_g}}}+\lambda _{2}\,{\it \_g}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_g}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}}}\,{\rm d}{\it \_g}}}\gamma _{1}-{{\rm e}^{\gamma _{1}\,{\it \_g}}}c_{1} \right ) }}}{d{\it \_g}}+{\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}},b_{2}\, \left ( -1+m \right ) \int ^{x}\!{{\rm e}^{b_{1}\, \left ( -1+m \right ) \int \!{{\rm e}^{\beta _{1}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_b}}}a_{1}}{\lambda _{1}}}}} \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_b}}}+\lambda _{2}\,{\it \_b}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_b}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}}}\,{\rm d}{\it \_b}+ \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_f}}}+\lambda _{2}\,{\it \_f}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_f}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}\beta _{2}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_b}}}a_{1}}{\lambda _{1}}}}}}}{d{\it \_b}}+{z}^{1-m}{{\rm e}^{b_{1}\,\int ^{x}\!{{\rm e}^{\beta _{1}\,{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,{\it \_b}}}a_{1}}{\lambda _{1}}}}} \left ( {\frac {1}{{y}^{k}} \left ( -{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a_{1}\, \left ( k-1 \right ) {{\rm e}^{\lambda _{1}\,{\it \_b}}}+\lambda _{2}\,{\it \_b}\,\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}{\it \_b}+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}a_{2}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\, \left ( k-1 \right ) +\lambda _{2}\,x\lambda _{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}} \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}}}{d{\it \_b}} \left ( -1+m \right ) }} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma _{1}\,x}}c_{1}}{\gamma _{1}}}}}\]
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Added Jan 19, 2020.
Problem Chapter 9.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 e^{\beta y} w_x + a_2 e^{sigma x} w_y + (b_1 x^n e^{\mu y}+b_2 y^m e^{\nu x+\lambda z}) w_z = c_1 w + c_2 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Exp[beta*y]*D[w[x,y,z],x]+ a2*Exp[sigma*x]*D[w[x,y,z],y]+(b1*x^n*Exp[mu*y]+b2*Exp[nu*x+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]];
Failed
Maple ✓
restart; local gamma; pde := a__1*exp(beta*y)*diff(w(x,y,z),x)+ a__1*exp(sigma*x)*diff(w(x,y,z),y)+ (b__1*x^n*exp(mu*y)+b__2*exp(nu*x+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'));
\[w \left ( x,y,z \right ) ={\frac {1}{c_{1}} \left ( {{\rm e}^{\sigma \,x}} \right ) ^{{\frac {c_{1}}{a_{1}\, \left ( {{\rm e}^{\beta \,y}}\sigma -{{\rm e}^{\sigma \,x}}\beta \right ) }}} \left ( \left ( {\frac {{{\rm e}^{\beta \,y}}\sigma }{\beta }} \right ) ^{{\frac {c_{1}}{a_{1}\, \left ( {{\rm e}^{\sigma \,x}}\beta -{{\rm e}^{\beta \,y}}\sigma \right ) }}}{\it \_F1} \left ( {\frac {{{\rm e}^{\beta \,y}}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\sigma \,\beta }},{\frac {1}{a_{1}\,\beta \,\lambda } \left ( -\int ^{x}\!{{{\rm e}^{{\frac {1}{a_{1}\,\beta } \left ( b_{1}\,\lambda \,\sigma \,\int \!{{{\it \_f}}^{n} \left ( -{\frac {-{{\rm e}^{\beta \,y}}\sigma +\beta \, \left ( {{\rm e}^{\sigma \,x}}-{{\rm e}^{\sigma \,{\it \_f}}} \right ) }{\sigma }} \right ) ^{{\frac {\mu }{\beta }}} \left ( {\frac {{{\rm e}^{\beta \,y}}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\beta }}+{{\rm e}^{\sigma \,{\it \_f}}} \right ) ^{-1}}\,{\rm d}{\it \_f}+\nu \,{\it \_f}\,a_{1}\,\beta \right ) }}} \left ( {\frac {{{\rm e}^{\beta \,y}}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\beta }}+{{\rm e}^{\sigma \,{\it \_f}}} \right ) ^{-1}}{d{\it \_f}}b_{2}\,\lambda \,\sigma -{{\rm e}^{{\frac {\lambda }{a_{1}\,\beta } \left ( \int ^{x}\!{{{\it \_a}}^{n} \left ( -{\frac {-{{\rm e}^{\beta \,y}}\sigma +\beta \, \left ( {{\rm e}^{\sigma \,x}}-{{\rm e}^{\sigma \,{\it \_f}}} \right ) }{\sigma }} \right ) ^{{\frac {\mu }{\beta }}} \left ( {\frac {{{\rm e}^{\beta \,y}}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\beta }}+{{\rm e}^{\sigma \,{\it \_f}}} \right ) ^{-1}}{d{\it \_f}}b_{1}\,\sigma -za_{1}\,\beta \right ) }}}a_{1}\,\beta \right ) } \right ) c_{1}-{{\rm e}^{{\frac {c_{1}\,\sigma \,x}{a_{1}\, \left ( {{\rm e}^{\sigma \,x}}\beta -{{\rm e}^{\beta \,y}}\sigma \right ) }}}}c_{2} \right ) }\]
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