#### 6.9.6 3.2

6.9.6.1 [1962] Problem 1
6.9.6.2 [1963] Problem 2
6.9.6.3 [1964] Problem 3
6.9.6.4 [1965] Problem 4
6.9.6.5 [1966] Problem 5
6.9.6.6 [1967] Problem 6
6.9.6.7 [1968] Problem 7
6.9.6.8 [1969] Problem 8
6.9.6.9 [1970] Problem 9
6.9.6.10 [1971] Problem 10

##### 6.9.6.1 [1962] Problem 1

problem number 1962

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'));



$w \left ( x,y,z \right ) = \left ( \int \!k{x}^{n}{{\rm e}^{-{\frac {{{\rm e}^{\beta \,x}}c}{\beta }}}}\,{\rm d}x+{\it \_F1} \left ( -ax+y,-bx+z \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\beta \,x}}c}{\beta }}}}$

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##### 6.9.6.2 [1963] Problem 2

problem number 1963

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'));



$w \left ( x,y,z \right ) = \left ( \int \!s{x}^{k}{{\rm e}^{-{\frac {{{\rm e}^{\gamma \,x}}c}{\gamma }}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {-xa{x}^{n}+y \left ( n+1 \right ) }{n+1}},{\frac {-b{{\rm e}^{\lambda \,x}}+z\lambda }{\lambda }} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma \,x}}c}{\gamma }}}}$

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##### 6.9.6.3 [1964] Problem 3

problem number 1964

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'));



$w \left ( x,y,z \right ) ={\frac {{{\rm e}^{ax}}}{a-\gamma } \left ( \left ( a-\gamma \right ) {\it \_F1} \left ( {\frac {-b{{\rm e}^{\beta \,x}}+y\beta }{\beta }},-\int ^{x}\!c \left ( {\frac {b{{\rm e}^{\beta \,{\it \_a}}}-b{{\rm e}^{\beta \,x}}+y\beta }{\beta }} \right ) ^{n}{d{\it \_a}}+z \right ) -s{{\rm e}^{-x \left ( a-\gamma \right ) }} \right ) }$

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##### 6.9.6.4 [1965] Problem 4

problem number 1965

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'));



$w \left ( x,y,z \right ) ={\frac {{{\rm e}^{c_{1}\,x}}}{c_{1}-\gamma } \left ( \left ( c_{1}-\gamma \right ) {\it \_F1} \left ( {\frac {{{\rm e}^{a_{1}\, \left ( k-1 \right ) x}} \left ( a_{2}\, \left ( -1+a_{1}\, \left ( k-1 \right ) x \right ) {y}^{k}+y{a_{1}}^{2} \left ( k-1 \right ) \right ) }{{y}^{k} \left ( k-1 \right ) {a_{1}}^{2}}},{\frac {1}{\lambda } \left ( -b_{2}\,\int ^{x}\!{{\rm e}^{ \left ( {\frac { \left ( a_{2}\, \left ( -1+a_{1}\, \left ( k-1 \right ) x \right ) {y}^{k}+y{a_{1}}^{2} \left ( k-1 \right ) \right ) {{\rm e}^{a_{1}\, \left ( x-{\it \_a} \right ) \left ( k-1 \right ) }}- \left ( -1+a_{1}\, \left ( k-1 \right ) {\it \_a} \right ) {y}^{k}a_{2}}{{y}^{k} \left ( k-1 \right ) {a_{1}}^{2}}} \right ) ^{- \left ( k-1 \right ) ^{-1}}\beta +1/2\,\lambda \,b_{1}\,{{\it \_a}}^{2}}}{d{\it \_a}}\lambda -{{\rm e}^{1/2\,\lambda \, \left ( b_{1}\,{x}^{2}-2\,z \right ) }} \right ) } \right ) -c_{2}\,{{\rm e}^{-x \left ( c_{1}-\gamma \right ) }} \right ) }$

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##### 6.9.6.5 [1966] Problem 5

problem number 1966

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]];



$\left \{\left \{w(x,y,z)\to e^{\text {c1} x} c_1\left (-\frac {\frac {\sqrt {2 \pi } \text {a2} \sqrt {\lambda } \text {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )}{\text {a1}^{3/2}}+\frac {2 e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a1}}}{2 \text {a2} \lambda ^2},(k-1) \int _1^x\text {b2} \exp \left (\frac {1}{2} \text {a1} \beta K[1]^2+\text {b1} (k-1) K[1]-\frac {\beta \left (\log (\text {a1})+\log (\text {a2})+2 \log (\lambda )+\log \left (\frac {\sqrt {\lambda } \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )+\frac {2 \sqrt {\text {a1}} e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a2}}-\sqrt {\lambda } \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } K[1]}{\sqrt {2}}\right )}{2 \text {a1}^{3/2} \lambda ^2}\right )\right )}{\lambda }\right )dK[1]+z^{1-k} e^{\text {b1} (k-1) x}\right )-\frac {\text {c2}}{\text {c1}}\right \}\right \}$

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'));



$w \left ( x,y,z \right ) ={\frac {1}{c_{1}} \left ( {{\rm e}^{c_{1}\,x}}{\it \_F1} \left ( 1/2\,{\frac {\sqrt {2}\sqrt {\pi }\sqrt {-\lambda \,a_{1}}\erf \left ( 1/2\,\sqrt {2}\sqrt {-\lambda \,a_{1}}x \right ) a_{2}\,\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}+2\,a_{1}}{\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}\lambda \,a_{1}}},b_{2}\,{2}^{1/2\,{\frac {\beta }{\lambda }}}{\pi }^{-1/2\,{\frac {\beta }{\lambda }}} \left ( k-1 \right ) \int ^{x}\!{{\rm e}^{1/2\,\beta \,a_{1}\,{{\it \_a}}^{2}+b_{1}\, \left ( k-1 \right ) {\it \_a}}}{\pi }^{{\frac {\beta }{\lambda }}} \left ( -4\,{{a_{1}}^{3}\lambda \left ( {\frac { \left ( \sqrt {2}\sqrt {\pi }\sqrt {-\lambda \,a_{1}}\erf \left ( 1/2\,\sqrt {2}\sqrt {-\lambda \,a_{1}}x \right ) a_{2}\,\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}+2\,a_{1} \right ) \sqrt {-2\,\lambda \,a_{1}\,\pi }}{\sqrt {{{\rm e}^{-\lambda \, \left ( a_{1}\,{x}^{2}-2\,y \right ) }}}}}+2\,a_{2}\,\erf \left ( 1/2\,\sqrt {-2\,\lambda \,a_{1}}{\it \_a} \right ) \pi \,a_{1}\,\lambda \right ) ^{-2}} \right ) ^{1/2\,{\frac {\beta }{\lambda }}}{d{\it \_a}}+{z}^{1-k}{{\rm e}^{b_{1}\, \left ( k-1 \right ) x}} \right ) c_{1}-c_{2} \right ) }$

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##### 6.9.6.6 [1967] Problem 6

problem number 1967

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'));



$w \left ( x,y,z \right ) ={\frac {1}{c_{1}} \left ( {{\rm e}^{c_{1}\,x}}{\it \_F1} \left ( {\frac {1}{\mu \,\lambda } \left ( a_{2}\,\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,x}}}{\mu }} \right ) \lambda -{{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\mu \,x}}-\mu \,y \right ) }{\mu }}}}\mu \right ) },{\frac {1}{\beta } \left ( -\int ^{x}\!{{\rm e}^{b_{1}\,\beta \,\int \!{{\rm e}^{{\frac {a_{1}\,\nu \,{{\rm e}^{\mu \,{\it \_f}}}}{\mu }}}} \left ( {\frac {1}{\mu } \left ( {{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\mu \,x}}-\mu \,y \right ) }{\mu }}}}\mu -a_{2}\,\lambda \, \left ( \Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,x}}}{\mu }} \right ) -\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,{\it \_f}}}}{\mu }} \right ) \right ) \right ) } \right ) ^{-{\frac {\nu }{\lambda }}}\,{\rm d}{\it \_f}}}{d{\it \_f}}b_{2}\,\beta -{{\rm e}^{\beta \, \left ( \int ^{x}\!{{\rm e}^{{\frac {\nu \,a_{1}\,{{\rm e}^{\mu \,{\it \_b}}}}{\mu }}}} \left ( {\frac {1}{\mu } \left ( {{\rm e}^{{\frac {\lambda \, \left ( a_{1}\,{{\rm e}^{\mu \,x}}-\mu \,y \right ) }{\mu }}}}\mu -a_{2}\,\lambda \, \left ( \Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,x}}}{\mu }} \right ) -\Ei \left ( 1,-{\frac {\lambda \,a_{1}\,{{\rm e}^{\mu \,{\it \_b}}}}{\mu }} \right ) \right ) \right ) } \right ) ^{-{\frac {\nu }{\lambda }}}{d{\it \_b}}b_{1}-z \right ) }} \right ) } \right ) c_{1}-c_{2} \right ) }$

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##### 6.9.6.7 [1968] Problem 7

problem number 1968

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]];



$\left \{\left \{w(x,y,z)\to e^{\frac {\text {c1} e^{\text {gamma1} x}}{\text {gamma1}}} \left (c_1\left (y e^{-\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}}-\int _1^x\text {a2} e^{\text {lambda2} K[1]-\frac {\text {a1} e^{\text {lambda1} K[1]}}{\text {lambda1}}}dK[1],z e^{-\frac {\text {b1} e^{\text {beta1} x}}{\text {beta1}}}-\int _1^x\text {b2} e^{\text {beta2} K[2]-\frac {\text {b1} e^{\text {beta1} K[2]}}{\text {beta1}}}dK[2]\right )+\int _1^x\text {c2} e^{\text {gamma2} K[3]-\frac {\text {c1} e^{\text {gamma1} K[3]}}{\text {gamma1}}}dK[3]\right )\right \}\right \}$

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'));



$w \left ( x,y,z \right ) = \left ( \int \!c_{2}\,{{\rm e}^{{\frac {\gamma _{2}\,x\gamma _{1}-{{\rm e}^{\gamma _{1}\,x}}c_{1}}{\gamma _{1}}}}}\,{\rm d}x+{\it \_F1} \left ( -a_{2}\,\int \!{{\rm e}^{{\frac {\lambda _{2}\,x\lambda _{1}-{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}}\,{\rm d}x+y{{\rm e}^{-{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}},{\frac {z\beta _{1}\,\beta _{2}-b_{1}\,{{\rm e}^{\beta _{1}\,x}}\beta _{2}-b_{2}\,{{\rm e}^{\beta _{2}\,x}}\beta _{1}}{\beta _{1}\,\beta _{2}}} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma _{1}\,x}}c_{1}}{\gamma _{1}}}}}$

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##### 6.9.6.8 [1969] Problem 8

problem number 1969

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]];



$\left \{\left \{w(x,y,z)\to e^{\frac {\text {c1} e^{\text {gamma1} x}}{\text {gamma1}}} \left (c_1\left ((k-1) \int _1^x\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {b1} e^{\text {beta1} K[2]} (m-1)}{\text {beta1}}+\text {beta2} K[2]}dK[2]+z^{1-m} e^{\frac {\text {b1} (m-1) e^{\text {beta1} x}}{\text {beta1}}}\right )+\int _1^x\text {c2} \exp \left (\text {gamma2} \left (e^{-\frac {\text {a1} \left (e^{\text {lambda1} K[3]} (k-1)+e^{\text {lambda1} x}\right )}{\text {lambda1}}} y^{-k} \left (e^{\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}} (k-1) \int _1^x\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1] y^k-e^{\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}} (k-1) \int _1^{K[3]}\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1] y^k+e^{\frac {\text {a1} e^{\text {lambda1} x} k}{\text {lambda1}}} y\right )\right ){}^{\frac {1}{1-k}}-\frac {\text {c1} e^{\text {gamma1} K[3]}}{\text {gamma1}}\right )dK[3]\right )\right \}\right \}$

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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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 ( m-1 \right ) \int \!{{\rm e}^{{\frac {b_{1}\,{{\rm e}^{\beta _{1}\,x}} \left ( m-1 \right ) +\beta _{2}\,x\beta _{1}}{\beta _{1}}}}}\,{\rm d}x+{{\rm e}^{{\frac {b_{1}\,{{\rm e}^{\beta _{1}\,x}}m}{\beta _{1}}}}}z \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}}}}}$

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##### 6.9.6.9 [1970] Problem 9

problem number 1970

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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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 ( m-1 \right ) }}+ \left ( \int ^{x}\!{{\rm e}^{b_{1}\, \left ( m-1 \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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 ( m-1 \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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 ) \left ( m-1 \right ) b_{2} \right ) ^{- \left ( m-1 \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \right ) \left ( {{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}}{\lambda _{1}}}}} \right ) ^{-1}},b_{2}\, \left ( m-1 \right ) \int ^{x}\!{{\rm e}^{b_{1}\, \left ( m-1 \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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+{{\rm e}^{{\frac {{{\rm e}^{\lambda _{1}\,x}}a_{1}\,k}{\lambda _{1}}}}}y \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 ( m-1 \right ) }} \right ) \right ) {{\rm e}^{{\frac {{{\rm e}^{\gamma _{1}\,x}}c_{1}}{\gamma _{1}}}}}$

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##### 6.9.6.10 [1971] Problem 10

problem number 1971

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'));



$w \left ( x,y,z \right ) ={\frac {1}{c_{1}} \left ( {{\rm e}^{\sigma \,x}} \right ) ^{{\frac {c_{1}}{a_{1}\, \left ( {{\rm e}^{y\beta }}\sigma -{{\rm e}^{\sigma \,x}}\beta \right ) }}} \left ( \left ( {\frac {{{\rm e}^{y\beta }}\sigma }{\beta }} \right ) ^{{\frac {c_{1}}{a_{1}\, \left ( {{\rm e}^{\sigma \,x}}\beta -{{\rm e}^{y\beta }}\sigma \right ) }}}{\it \_F1} \left ( {\frac {{{\rm e}^{y\beta }}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\sigma \,\beta }},{\frac {1}{a_{1}\,\beta \,\lambda } \left ( -\int ^{x}\!{{\rm e}^{{\frac {1}{a_{1}\,\beta } \left ( \lambda \,\sigma \,b_{1}\,\int \!{{{\it \_b}}^{n} \left ( -{\frac {-{{\rm e}^{y\beta }}\sigma +\beta \, \left ( {{\rm e}^{\sigma \,x}}-{{\rm e}^{\sigma \,{\it \_b}}} \right ) }{\sigma }} \right ) ^{{\frac {\mu }{\beta }}} \left ( {\frac {{{\rm e}^{y\beta }}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\beta }}+{{\rm e}^{\sigma \,{\it \_b}}} \right ) ^{-1}}\,{\rm d}{\it \_b}+\nu \,{\it \_b}\,a_{1}\,\beta \right ) }}} \left ( {\frac {{{\rm e}^{y\beta }}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\beta }}+{{\rm e}^{\sigma \,{\it \_b}}} \right ) ^{-1}{d{\it \_b}}b_{2}\,\lambda \,\sigma -{{\rm e}^{{\frac {\lambda }{a_{1}\,\beta } \left ( \int ^{x}\!{{{\it \_a}}^{n} \left ( -{\frac {-{{\rm e}^{y\beta }}\sigma +\beta \, \left ( {{\rm e}^{\sigma \,x}}-{{\rm e}^{\sigma \,{\it \_b}}} \right ) }{\sigma }} \right ) ^{{\frac {\mu }{\beta }}} \left ( {\frac {{{\rm e}^{y\beta }}\sigma -{{\rm e}^{\sigma \,x}}\beta }{\beta }}+{{\rm e}^{\sigma \,{\it \_b}}} \right ) ^{-1}}{d{\it \_b}}b_{1}\,\sigma -za_{1}\,\beta \right ) }}}a_{1}\,\beta \right ) } \right ) c_{1}-{{\rm e}^{{\frac {c_{1}\,\sigma \,x}{a_{1}\, \left ( {{\rm e}^{\sigma \,x}}\beta -{{\rm e}^{y\beta }}\sigma \right ) }}}}c_{2} \right ) }$

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