Added June 27, 2019.
Problem Chapter 8.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c w_z = (\alpha x+\beta y+\gamma z+ \delta ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right ) \exp \left (\frac {x (a (\alpha x+2 \beta y+2 \delta +2 \gamma z)-x (b \beta +c \gamma ))}{2 a^2}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*diff(w(x,y,z),z)= (alpha*x+beta*y+gamma*z+delta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {za-cx}{a}} \right ) {{\rm e}^{{\frac {x}{{a}^{2}} \left ( \left ( \beta \,y+\gamma \,z+{\frac {\alpha \,x}{2}}+\delta \right ) a-{\frac {x \left ( \beta \,b+c\gamma \right ) }{2}} \right ) }}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a z w_y + b y w_z = (c x+s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*z*D[w[x, y,z], y] + b*y*D[w[x,y,z],z]== (c*x+s)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^2}{2}+s x} c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y)+ b*y*diff(w(x,y,z),z)= (c*x+s)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{z}^{2}a-b{y}^{2}}{a}},-{ \left ( -x\sqrt {ba}+\ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) \right ) {\frac {1}{\sqrt {ba}}}} \right ) {{\rm e}^{\int ^{y}\!-{ \left ( \left ( -cx-s \right ) \sqrt {ba}+c \left ( \ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) -\ln \left ( { \left ( {\it \_a}\,ab+\sqrt { \left ( {z}^{2}a+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) a}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) \right ) \right ) {\frac {1}{\sqrt { \left ( {z}^{2}a+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) a}}}{\frac {1}{\sqrt {ba}}}}{d{\it \_a}}}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x+a_0) w_y + (b_1 x+b_0) w_z = (\alpha x+\beta y+\gamma z+\delta ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*x+a0)*D[w[x, y,z], y] + (b1*x+b0)*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\text {a0} x-\frac {\text {a1} x^2}{2}+y,-\text {b0} x-\frac {\text {b1} x^2}{2}+z\right ) \exp \left (\frac {1}{6} x \left (-3 \text {a0} \beta x-2 \text {a1} \beta x^2+3 \alpha x-3 \text {b0} \gamma x-2 \text {b1} \gamma x^2+6 \beta y+6 \delta +6 \gamma z\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a1*x+a0)*diff(w(x,y,z),y)+ (b1*x+b0)*diff(w(x,y,z),z)= (alpha*x+beta*y+gamma*z+delta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac {{\it a1}\,{x}^{2}}{2}}-{\it a0}\,x+y,-{\frac {{\it b1}\,{x}^{2}}{2}}-{\it b0}\,x+z \right ) {{\rm e}^{-{\frac {x}{3} \left ( \left ( {\it b1}\,{x}^{2}+{\frac {3\,{\it b0}\,x}{2}}-3\,z \right ) \gamma + \left ( {\it a1}\,{x}^{2}+{\frac {3\,{\it a0}\,x}{2}}-3\,y \right ) \beta -{\frac {3\,\alpha \,x}{2}}-3\,\delta \right ) }}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_2 y+a_1 x+a_0) w_y + (b_2 y+b_1 x+b_0) w_z = (c_2 y+c_1 z+c_0 x+s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a2*y+a1*x+a0)*D[w[x, y,z], y] + (b2*y+b1*x+b0)*D[w[x,y,z],z]== (c2*y+c1*z+c0*x+s)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\text {a2} x} (\text {a2} (\text {a0}+\text {a2} y)+\text {a1} \text {a2} x+\text {a1})}{\text {a2}^2},\frac {e^{-\text {a2} x} \left (\text {a2} \left (2 \text {a0} \text {b2} \left (\text {a2} x e^{\text {a2} x}+1\right )-\text {a2} \left (\text {a2} e^{\text {a2} x} \left (2 \text {b0} x+\text {b1} x^2-2 z\right )+2 \text {b2} y \left (e^{\text {a2} x}-1\right )\right )\right )+\text {a1} \text {b2} \left (\text {a2}^2 x^2 e^{\text {a2} x}+2 \text {a2} x+2\right )\right )}{2 \text {a2}^3}\right ) \exp \left (\frac {\text {a2} \left (3 \text {a0} \text {b2} \text {c1} \left (\text {a2}^2 x^2-2 \text {a2} x+2\right )-6 \text {a0} \text {a2} \text {c2} (\text {a2} x-1)+\text {a2}^3 x \left (-3 \text {b0} \text {c1} x-2 \text {b1} \text {c1} x^2+3 \text {c0} x+6 \text {c1} z+6 s\right )+6 \text {a2}^2 y (\text {c2}-\text {b2} \text {c1} x)+6 \text {a2} \text {b2} \text {c1} y\right )+\text {a1} \left (\text {b2} \text {c1} \left (2 \text {a2}^3 x^3-3 \text {a2}^2 x^2+6\right )-3 \text {a2} \text {c2} \left (\text {a2}^2 x^2-2\right )\right )}{6 \text {a2}^4}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a2*y+a1*x+a0)*diff(w(x,y,z),y)+ (b2*y+b1*x+b0)*diff(w(x,y,z),z)= (c2*y+c1*z+c0*x+s)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{{\rm e}^{-{\it a2}\,x}} \left ( y{{\it a2}}^{2}+{\it a2}\, \left ( {\it a1}\,x+{\it a0} \right ) +{\it a1} \right ) }{{{\it a2}}^{2}}},{\frac { \left ( -{\it b1}\,{x}^{2}-2\,{\it b0}\,x+2\,z \right ) {{\it a2}}^{3}+{\it b2}\, \left ( {\it a1}\,{x}^{2}+2\,{\it a0}\,x-2\,y \right ) {{\it a2}}^{2}-2\,{\it a0}\,{\it a2}\,{\it b2}-2\,{\it a1}\,{\it b2}}{2\,{{\it a2}}^{3}}} \right ) {{\rm e}^{{\frac {1}{6\,{{\it a2}}^{4}} \left ( \left ( \left ( 2\,{x}^{3}{\it a1}\,{{\it a2}}^{3}+ \left ( 3\,{\it a0}\,{{\it a2}}^{3}-3\,{\it a1}\,{{\it a2}}^{2} \right ) {x}^{2}+ \left ( -6\,{{\it a2}}^{3}y-6\,{\it a0}\,{{\it a2}}^{2} \right ) x+6\,y{{\it a2}}^{2}+6\,{\it a0}\,{\it a2}+6\,{\it a1} \right ) {\it b2}-2\,x{{\it a2}}^{4} \left ( {\it b1}\,{x}^{2}+3/2\,{\it b0}\,x-3\,z \right ) \right ) {\it c1}+6\, \left ( 1/2\,{{\it a2}}^{2} \left ( -{\it c2}\,{\it a1}+{\it a2}\,{\it c0} \right ) {x}^{2}+ \left ( -{\it a0}\,{{\it a2}}^{2}{\it c2}+{{\it a2}}^{3}s \right ) x+{\it c2}\, \left ( y{{\it a2}}^{2}+{\it a0}\,{\it a2}+{\it a1} \right ) \right ) {\it a2} \right ) }}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y+k_1 x+k_0) w_y + (b z+s_1 x+s_0) w_z = (c_1 x+c_0) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+k1*x+k0)*D[w[x, y,z], y] + (b*z+s1*x+s0)*D[w[x,y,z],z]== (c1*x+c0)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\text {c0} x+\frac {\text {c1} x^2}{2}} c_1\left (\frac {e^{-a x} \left (a^2 y+a (\text {k0}+\text {k1} x)+\text {k1}\right )}{a^2},\frac {e^{-b x} \left (b^2 z+b (\text {s0}+\text {s1} x)+\text {s1}\right )}{b^2}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a*y+k1*x+k0)*diff(w(x,y,z),y)+(b*z+s1*x+s0)*diff(w(x,y,z),z)= (c1*x+c0)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{{\rm e}^{-ax}} \left ( y{a}^{2}+a \left ( {\it k1}\,x+{\it k0} \right ) +{\it k1} \right ) }{{a}^{2}}},{\frac {{{\rm e}^{-xb}} \left ( z{b}^{2}+b \left ( {\it s1}\,x+{\it s0} \right ) +{\it s1} \right ) }{{b}^{2}}} \right ) {{\rm e}^{{\frac {x \left ( {\it c1}\,x+2\,{\it c0} \right ) }{2}}}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + c z w_z = (\alpha x+\beta y+\gamma z+\delta ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] + c*z*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+beta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x^{\frac {\beta }{a}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right ) e^{\frac {\alpha x}{a}+\frac {\beta y}{b}+\frac {\gamma z}{c}}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+c*z*diff(w(x,y,z),z)= (alpha*x+beta*y+gamma*z+beta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={x}^{{\frac {\beta }{a}}}{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) {{\rm e}^{{\frac { \left ( ya\beta +\alpha \,xb \right ) c+z\gamma \,ab}{abc}}}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + a z w_y + b y w_z = c w \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + a*z*D[w[x, y,z], y] + b*y*D[w[x,y,z],z]== c*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x^c c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := x*diff(w(x,y,z),x)+ a*z*y*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= c*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{z}^{2}a-2\,by}{a}},x{{\rm e}^{2\,{\frac {1}{a}\arctanh \left ( {z{\frac {1}{\sqrt {{\frac {{z}^{2}a-2\,by}{a}}}}}} \right ) {\frac {1}{\sqrt {{\frac {{z}^{2}a-2\,by}{a}}}}}}}} \right ) {{\rm e}^{-2\,{\frac {c}{a}\arctanh \left ( {\frac {\sqrt {{a}^{2}{z}^{2}}}{a}{\frac {1}{\sqrt {{\frac {{z}^{2}a-2\,by}{a}}}}}} \right ) {\frac {1}{\sqrt {{\frac {{z}^{2}a-2\,by}{a}}}}}}}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a b x w_x + b(a y + b z) w_y + a(a y -b z) w_z = c w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*b*x*D[w[x, y,z], x] + b*(a*y+b*z)*D[w[x, y,z], y] + a*(a*y-b*z)*D[w[x,y,z],z]== c*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*b*x*diff(w(x,y,z),x)+ b*(a*y+b*z)*diff(w(x,y,z),y)+a*(a*y-b*z)*diff(w(x,y,z),z)= c*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}},x \left ( { \left ( {\frac {y{a}^{2}\sqrt {2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+ \left ( {ay{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}+{bz{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}} \right ) \sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}} \right ) ^{-{\frac {a\sqrt {2}}{2}{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}} \right ) \left ( {\frac {y{a}^{2}\sqrt {2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}+{ay{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}+{bz{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}} \right ) ^{{\frac {c\sqrt {2}}{2\,b}{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}}\]
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Added June 27, 2019.
Problem Chapter 8.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (a_1 x +a_0) w_x + (b_1 y + b_0) w_y + (c_1 z +c_0) w_z = (\alpha x+\beta y+\gamma z+\delta ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a1*x+a0)*D[w[x, y,z], x] + (b1*y+b0)*D[w[x, y,z], y] +(c1*z+c0)*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to (\text {a0}+\text {a1} x)^{-\frac {\text {a0} \alpha \text {b1} \text {c1}+\text {a1} \text {b0} \beta \text {c1}+\text {a1} \text {b1} \text {c0} \gamma -\text {a1} \text {b1} \text {c1} \delta }{\text {a1}^2 \text {b1} \text {c1}}} c_1\left (\frac {(\text {b0}+\text {b1} y) (\text {a0}+\text {a1} x)^{-\frac {\text {b1}}{\text {a1}}}}{\text {b1}},\frac {(\text {c0}+\text {c1} z) (\text {a0}+\text {a1} x)^{-\frac {\text {c1}}{\text {a1}}}}{\text {c1}}\right ) \exp \left (\frac {\alpha x}{\text {a1}}+\frac {\beta (\text {b0}+\text {b1} y)}{\text {b1}^2}+\frac {\gamma (\text {c0}+\text {c1} z)}{\text {c1}^2}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := (a1*x+a0)*diff(w(x,y,z),x)+(b1*y+b0)*diff(w(x,y,z),y)+(c1*z+c0)*diff(w(x,y,z),z)= (alpha*x+beta*y+gamma*z+delta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{\it b1}\,y+{\it b0}}{{\it b1}} \left ( {\it a1}\,x+{\it a0} \right ) ^{-{\frac {{\it b1}}{{\it a1}}}}},{\frac {{\it c1}\,z+{\it c0}}{{\it c1}} \left ( {\it a1}\,x+{\it a0} \right ) ^{-{\frac {{\it c1}}{{\it a1}}}}} \right ) \left ( {\it a1}\,x+{\it a0} \right ) ^{-{\frac {{\it a0}\,\alpha }{{{\it a1}}^{2}}}-{\frac {{\it b0}\,\beta }{{\it a1}\,{\it b1}}}-{\frac {{\it c0}\,\gamma }{{\it a1}\,{\it c1}}}+{\frac {\delta }{{\it a1}}}}{{\rm e}^{{\frac {\alpha \,x}{{\it a1}}}+{\frac {\beta \, \left ( {\it b1}\,y+{\it b0} \right ) }{{{\it b1}}^{2}}}+{\frac { \left ( {\it c1}\,z+{\it c0} \right ) \gamma }{{{\it c1}}^{2}}}}}\]
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