Added April 2, 2019.
Problem Chapter 5.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda x} y+ b x^n) w_y = c w + k e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y+b*x^n)*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (y e^{-\frac {a e^{\lambda x}}{\lambda }}-\int _1^xb e^{-\frac {a e^{\lambda K[1]}}{\lambda }} K[1]^ndK[1]\right )\right )}{c-\gamma }\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*x^n)*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {k \,{\mathrm e}^{\left (-c +\gamma \right ) x}}{-c +\gamma }+\textit {\_F1} \left (-b \left (\int x^{n} {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}d x \right )+y \,{\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}\right )\right ) {\mathrm e}^{c x}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda x} y+ b e^{\beta x}) w_y = c w + k e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y+b*Exp[beta*x])*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (y e^{-\frac {a e^{\lambda x}}{\lambda }}-\int _1^xb e^{\beta K[1]-\frac {a e^{\lambda K[1]}}{\lambda }}dK[1]\right )\right )}{c-\gamma }\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*exp(beta*x))*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = -\frac {\left (k \,{\mathrm e}^{\left (-c +\gamma \right ) x}+\left (-c +\gamma \right ) \textit {\_F1} \left (-b \left (\int {\mathrm e}^{\frac {\beta \lambda x -a \,{\mathrm e}^{\lambda x}}{\lambda }}d x \right )+y \,{\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}\right )\right ) {\mathrm e}^{c x}}{c -\gamma }\]
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Added April 2, 2019.
Problem Chapter 5.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda x} y+ b e^{\beta x}) w_y = c w + k x^n \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y+b*Exp[beta*x])*D[w[x, y], y] == c*w[x,y]+k*x^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{c x} \left (-\frac {k x^n (c x)^{-n} \operatorname {Gamma}(n+1,c x)}{c}+c_1\left (y e^{-\frac {a e^{\lambda x}}{\lambda }}-\int _1^xb e^{\beta K[1]-\frac {a e^{\lambda K[1]}}{\lambda }}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*exp(beta*x))*diff(w(x,y),y) = c*w(x,y)+k*x^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (k \,x^{n} \left (c x \right )^{-\frac {n}{2}} \WhittakerM \left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, c x \right ) {\mathrm e}^{-\frac {c x}{2}}+\left (n +1\right ) c \textit {\_F1} \left (-b \left (\int {\mathrm e}^{\frac {\beta \lambda x -a \,{\mathrm e}^{\lambda x}}{\lambda }}d x \right )+y \,{\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}\right )\right ) {\mathrm e}^{c x}}{\left (n +1\right ) c}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda y} + b x^k) w_y = c w + k e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*y]+b*x^k)*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (\frac {a \lambda x \left (-\frac {b \lambda x^{k+1}}{k+1}\right )^{-\frac {1}{k+1}} \operatorname {Gamma}\left (\frac {1}{k+1},-\frac {b \lambda x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\lambda \left (-b x^{k+1}+k y+y\right )}{k+1}}}{a b k (k+1) \lambda ^2}\right )\right )}{c-\gamma }\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*y)+b*x^k)*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = -\frac {\left (k \,{\mathrm e}^{\left (-c +\gamma \right ) x}+\left (-c +\gamma \right ) \textit {\_F1} \left (\frac {\left (k +1\right ) \left (k +2\right )^{2} a \,x^{-k} \left (-\frac {b \lambda \,x^{k +1}}{k +1}\right )^{\frac {-k -2}{2 k +2}} \WhittakerM \left (\frac {k +2}{2 k +2}, \frac {2 k +3}{2 k +2}, -\frac {b \lambda \,x^{k +1}}{k +1}\right ) {\mathrm e}^{\frac {b \lambda \,x^{k +1}}{2 k +2}}-\left (b \lambda x +\left (-k -2\right ) x^{-k}\right ) \left (k +1\right )^{2} a \left (-\frac {b \lambda \,x^{k +1}}{k +1}\right )^{\frac {-k -2}{2 k +2}} \WhittakerM \left (-\frac {k}{2 k +2}, \frac {2 k +3}{2 k +2}, -\frac {b \lambda \,x^{k +1}}{k +1}\right ) {\mathrm e}^{\frac {b \lambda \,x^{k +1}}{2 k +2}}-2 \left (k +2\right ) \left (k +\frac {3}{2}\right ) b \,{\mathrm e}^{\frac {\left (b \,x^{k +1}-\left (k +1\right ) y \right ) \lambda }{k +1}}}{\left (2 k^{2}+7 k +6\right ) b \lambda }\right )\right ) {\mathrm e}^{c x}}{c -\gamma }\]
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Added April 2, 2019.
Problem Chapter 5.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x e^{\lambda x+\mu y} w + b e^{\nu x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Exp[lambda*x+mu*y]*w[x,y]+b*Exp[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {a x e^{\lambda x+\mu y}}{\lambda x+\mu y}} \left (\int _1^x\frac {b \exp \left (\nu K[1]-\frac {a e^{\left (\lambda +\frac {\mu y}{x}\right ) K[1]} x}{\lambda x+\mu y}\right )}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x* diff(w(x,y),x)+ y*diff(w(x,y),y) = a*x*exp(lambda*x+mu*y)*w(x,y)+k*exp(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {k \,{\mathrm e}^{-\frac {a x \,{\mathrm e}^{\textit {\_a} \lambda +\frac {\textit {\_a} \mu y}{x}}-\left (\lambda x +\mu y \right ) \textit {\_a} \nu }{\lambda x +\mu y}}}{\textit {\_a}}d \textit {\_a} +\textit {\_F1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x +\mu y}}{\lambda +\frac {\mu y}{x}}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = (a y e^{\lambda x}+ b x e^{\mu y}) w + c e^{\nu x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == (a*y*Exp[lambda*x]+b*x*Exp[mu*y])*w[x,y]+c*Exp[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {a y e^{\lambda x}}{\lambda x}+\frac {b x e^{\mu y}}{\mu y}} \left (\int _1^x\frac {c \exp \left (-\frac {b e^{\frac {\mu y K[1]}{x}} x}{\mu y}+\nu K[1]-\frac {a e^{\lambda K[1]} y}{\lambda x}\right )}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x* diff(w(x,y),x)+ y*diff(w(x,y),y) = (a*y*exp(lambda*x)+b*x*exp(mu*y))*w(x,y)+c*exp(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {c \,{\mathrm e}^{-\frac {\left (-\frac {\textit {\_a} \lambda \mu \nu y}{x}+\frac {a \mu \,y^{2} {\mathrm e}^{\textit {\_a} \lambda }}{x^{2}}+b \lambda \,{\mathrm e}^{\frac {\textit {\_a} \mu y}{x}}\right ) x}{\lambda \mu y}}}{\textit {\_a}}d \textit {\_a} +\textit {\_F1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{\frac {\left (\frac {a \mu \,y^{2} {\mathrm e}^{\lambda x}}{x^{2}}+b \lambda \,{\mathrm e}^{\mu y}\right ) x}{\lambda \mu y}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^k w_x + b e^{\lambda x} w_y = w + c e^{\beta x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y^k*D[w[x, y], x] + b*Exp[lambda*x]*D[w[x, y], y] == w[x,y]+c*Exp[beta*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (-\frac {(k+1) y^{k+1} \left (\left (y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k} \, _2F_1\left (1,\frac {1}{k+1};\frac {k+2}{k+1};\frac {a \lambda y^{k+1}}{a \lambda y^{k+1}-b e^{\lambda x} (k+1)}\right )}{a \lambda y^{k+1}-b (k+1) e^{\lambda x}}\right ) \left (\int _1^x\frac {c \exp \left (\frac {(k+1) \left (a \lambda y^{k+1}-b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)\right ) \, _2F_1\left (1,\frac {1}{k+1};\frac {k+2}{k+1};1-\frac {b e^{\lambda K[1]} (k+1)}{b e^{\lambda x} (k+1)-a \lambda y^{k+1}}\right ) \left (\left (y^{k+1}-\frac {b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a \lambda \left (a \lambda y^{k+1}-b e^{\lambda x} (k+1)\right )}+\beta K[1]\right ) \left (\left (y^{k+1}-\frac {b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b e^{\lambda x}}{a \lambda }\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*y^k* diff(w(x,y),x)+ b*exp(lambda*x)*diff(w(x,y),y) = w(x,y)+c*exp(beta*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {c \left (\left (\frac {a \lambda \,y^{k +1}+\left (k +1\right ) b \,{\mathrm e}^{\textit {\_b} \lambda }-\left (k +1\right ) b \,{\mathrm e}^{\lambda x}}{a \lambda }\right )^{\frac {1}{k +1}}\right )^{-k} {\mathrm e}^{\frac {\textit {\_b} a \beta -\left (\int \left (\left (\frac {a \lambda \,y^{k +1}+\left (k +1\right ) b \,{\mathrm e}^{\textit {\_b} \lambda }-\left (k +1\right ) b \,{\mathrm e}^{\lambda x}}{a \lambda }\right )^{\frac {1}{k +1}}\right )^{-k}d \textit {\_b} \right )}{a}}}{a}d \textit {\_b} +\textit {\_F1} \left (\frac {a \lambda y \,y^{k}-\left (k +1\right ) b \,{\mathrm e}^{\lambda x}}{a \lambda }\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\left (\frac {a \lambda \,y^{k +1}+\left (k +1\right ) b \,{\mathrm e}^{\textit {\_a} \lambda }-\left (k +1\right ) b \,{\mathrm e}^{\lambda x}}{a \lambda }\right )^{\frac {1}{k +1}}\right )^{-k}}{a}d \textit {\_a}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda x} w_x + b y w_y = w + c e^{\lambda x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*y*D[w[x, y], y] == w[x,y]+c*Exp[lambda*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {e^{-\frac {e^{-\lambda x}}{a \lambda }} \left (-c \text {Ei}\left (\frac {e^{-\lambda x}}{a \lambda }\right )+a \lambda c_1\left (y e^{\frac {b e^{-\lambda x}}{a \lambda }}\right )\right )}{a \lambda }\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*x)* diff(w(x,y),x)+ b*y*diff(w(x,y),y) = w(x,y)+c*exp(lambda*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (a \lambda \textit {\_F1} \left (y \,{\mathrm e}^{\frac {b \,{\mathrm e}^{-\lambda x}}{a \lambda }}\right )+c \Ei \left (1, -\frac {{\mathrm e}^{-\lambda x}}{a \lambda }\right )\right ) {\mathrm e}^{-\frac {{\mathrm e}^{-\lambda x}}{a \lambda }}}{a \lambda }\]
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Added April 2, 2019.
Problem Chapter 5.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda y} w_x + b x^k w_y = w + c e^{\beta x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*y]*D[w[x, y], x] + b*x^k*D[w[x, y], y] == w[x,y]+c*Exp[beta*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {(k+1) x \, _2F_1\left (1,\frac {1}{k+1};1+\frac {1}{k+1};\frac {b \lambda x^{k+1}}{b \lambda x^{k+1}-a e^{\lambda y} (k+1)}\right )}{a (k+1) e^{\lambda y}-b \lambda x^{k+1}}\right ) \left (\int _1^x\frac {c \exp \left (\left (\beta -\frac {(k+1) \, _2F_1\left (1,\frac {1}{k+1};1+\frac {1}{k+1};\frac {b \lambda K[1]^{k+1}}{b \lambda x^{k+1}-a e^{\lambda y} (k+1)}\right )}{a e^{\lambda y} (k+1)-b \lambda x^{k+1}}\right ) K[1]\right ) (k+1)}{a e^{\lambda y} (k+1)+b \lambda \left (K[1]^{k+1}-x^{k+1}\right )}dK[1]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b x^{k+1}}{a k+a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*y)* diff(w(x,y),x)+ b*x^k*diff(w(x,y),y) = w(x,y)+c*exp(beta*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (k +1\right ) c \,{\mathrm e}^{\frac {\textit {\_b} b \beta \lambda +\left (-k -1\right ) \left (\int \frac {b \lambda }{b \lambda \,\textit {\_b}^{k +1}-b \lambda \,x^{k +1}+\left (k +1\right ) a \,{\mathrm e}^{\lambda y}}d \textit {\_b} \right )}{b \lambda }}}{b \lambda \,\textit {\_b}^{k +1}-b \lambda \,x^{k +1}+\left (k +1\right ) a \,{\mathrm e}^{\lambda y}}d \textit {\_b} +\textit {\_F1} \left (\frac {-b \lambda \,x^{k +1}+\left (k +1\right ) a \,{\mathrm e}^{\lambda y}}{\left (k +1\right ) b \lambda }\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {k +1}{b \lambda \,\textit {\_a}^{k +1}-b \lambda \,x^{k +1}+\left (k +1\right ) a \,{\mathrm e}^{\lambda y}}d \textit {\_a}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda y} w_x + b e^{\beta x} w_y = w + c x^k \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*y]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == w[x,y]+c*x^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\beta x-\log \left (\frac {a \beta e^{\lambda y}}{\lambda }\right )}{a \beta e^{\lambda y}-b \lambda e^{\beta x}}\right ) \left (\int _1^x\frac {\beta c \exp \left (\frac {\log \left (\frac {a e^{\lambda y} \beta }{\lambda }+b \left (-e^{\beta x}+e^{\beta K[1]}\right )\right )-\beta K[1]}{a \beta e^{\lambda y}-b e^{\beta x} \lambda }\right ) K[1]^k}{a e^{\lambda y} \beta +b \left (-e^{\beta x}+e^{\beta K[1]}\right ) \lambda }dK[1]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b e^{\beta x}}{a \beta }\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*y)* diff(w(x,y),x)+ b*exp(beta*x)*diff(w(x,y),y) = w(x,y)+c*x^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\beta c \,\textit {\_a}^{k} \left ({\mathrm e}^{\textit {\_a} \beta }+\frac {a \beta \,{\mathrm e}^{\lambda y}-b \lambda \,{\mathrm e}^{\beta x}}{b \lambda }\right )^{\frac {-a \beta \,{\mathrm e}^{\lambda y}+b \lambda \,{\mathrm e}^{\beta x}+1}{a \beta \,{\mathrm e}^{\lambda y}-b \lambda \,{\mathrm e}^{\beta x}}} \left ({\mathrm e}^{\textit {\_a} \beta }\right )^{-\frac {1}{a \beta \,{\mathrm e}^{\lambda y}-b \lambda \,{\mathrm e}^{\beta x}}}}{b \lambda }d \textit {\_a} +\textit {\_F1} \left (\frac {a \beta \,{\mathrm e}^{\lambda y}-b \lambda \,{\mathrm e}^{\beta x}}{b \beta \lambda }\right )\right ) \left (\frac {a \beta \,{\mathrm e}^{\lambda y}}{b \lambda }\right )^{-\frac {1}{a \beta \,{\mathrm e}^{\lambda y}-b \lambda \,{\mathrm e}^{\beta x}}} \left ({\mathrm e}^{\beta x}\right )^{\frac {1}{a \beta \,{\mathrm e}^{\lambda y}-b \lambda \,{\mathrm e}^{\beta x}}}\]
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