6.5.6 3.2

6.5.6.1 [1239] Problem 1

problem number 1239

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'));
 

\[w \left (x , y\right ) = \frac {-k \,{\mathrm e}^{\gamma x}+{\mathrm e}^{c x} f_{1} \left (-b \int x^{n} {\mathrm e}^{-\frac {{\mathrm e}^{\lambda x} a}{\lambda }}d x +{\mathrm e}^{-\frac {{\mathrm e}^{\lambda x} a}{\lambda }} y \right ) \left (c -\gamma \right )}{c -\gamma }\]

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6.5.6.2 [1240] Problem 2

problem number 1240

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'));
 

\[w \left (x , y\right ) = \frac {{\mathrm e}^{c x} \left (c -\gamma \right ) f_{1} \left (-b \int {\mathrm e}^{\frac {\beta x \lambda -{\mathrm e}^{\lambda x} a}{\lambda }}d x +{\mathrm e}^{-\frac {{\mathrm e}^{\lambda x} a}{\lambda }} y \right )-k \,{\mathrm e}^{\gamma x}}{c -\gamma }\]

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6.5.6.3 [1241] Problem 3

problem number 1241

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]];
 

\[\left \{\left \{w(x,y)\to e^{c x} \left (-\frac {k x^n (c x)^{-n} \Gamma (n+1,c x)}{c}+c_1\left (y e^{-\frac {a e^{\lambda x}}{\lambda }}-\int _1^xb e^{\beta K[1]-\frac {a e^{\lambda K[1]}}{\lambda }}dK[1]\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {{\mathrm e}^{c x} c \left (n +1\right ) f_{1} \left ({\mathrm e}^{-\frac {{\mathrm e}^{\lambda x} a}{\lambda }} y -b \int {\mathrm e}^{\frac {\beta x \lambda -{\mathrm e}^{\lambda x} a}{\lambda }}d x \right )+\left (c x \right )^{-\frac {n}{2}} \operatorname {WhittakerM}\left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, c x \right ) x^{n} k \,{\mathrm e}^{\frac {c x}{2}}}{c \left (n +1\right )}\]

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6.5.6.4 [1242] Problem 4

problem number 1242

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]];
 

\[\left \{\left \{w(x,y)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (\frac {a \lambda x \left (-\frac {b \lambda x^{k+1}}{k+1}\right )^{-\frac {1}{k+1}} \Gamma \left (\frac {1}{k+1},-\frac {b \lambda x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\lambda \left (-b x^{k+1}+k y+y\right )}{k+1}}}{a b k (k+1) \lambda ^2}\right )\right )}{c-\gamma }\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {{\mathrm e}^{c x} \left (c -\gamma \right ) f_{1} \left (-\frac {\left (-{\mathrm e}^{\frac {x \lambda b \,x^{k}}{2 k +2}} a \left (k +1\right ) \left (k +2\right )^{2} \operatorname {WhittakerM}\left (\frac {k +2}{2 k +2}, \frac {2 k +3}{2 k +2}, -\frac {x \lambda b \,x^{k}}{k +1}\right )+{\mathrm e}^{\frac {x \lambda b \,x^{k}}{2 k +2}} a \left (k +1\right )^{2} \left (x \lambda b \,x^{k}-k -2\right ) \operatorname {WhittakerM}\left (-\frac {k}{2 k +2}, \frac {2 k +3}{2 k +2}, -\frac {x \lambda b \,x^{k}}{k +1}\right )+2 \,{\mathrm e}^{\frac {\left (b x \,x^{k}-y \left (k +1\right )\right ) \lambda }{k +1}} \left (k +\frac {3}{2}\right ) \left (-\frac {x \lambda b \,x^{k}}{k +1}\right )^{\frac {k +2}{2 k +2}} x^{k} \left (k +2\right ) b \right ) x^{-k} \left (-\frac {x \lambda b \,x^{k}}{k +1}\right )^{\frac {-k -2}{2 k +2}}}{b \lambda \left (2 k^{2}+7 k +6\right )}\right )-k \,{\mathrm e}^{\gamma x}}{c -\gamma }\]

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6.5.6.5 [1243] Problem 5

problem number 1243

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'));
 

\[w \left (x , y\right ) = \left (k \int _{}^{x}\frac {{\mathrm e}^{\frac {-a \,{\mathrm e}^{\frac {\textit {\_a} \left (\lambda x +\mu y \right )}{x}} x +\textit {\_a} \nu \left (\lambda x +\mu y \right )}{\lambda x +\mu y}}}{\textit {\_a}}d \textit {\_a} +f_{1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x +\mu y} x}{\lambda x +\mu y}}\]

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6.5.6.6 [1244] Problem 6

problem number 1244

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'));
 

\[w \left (x , y\right ) = \left (c \int _{}^{x}\frac {{\mathrm e}^{\frac {-b \,x^{2} {\mathrm e}^{\frac {\mu y \textit {\_a}}{x}} \lambda -y \mu \left (-x \textit {\_a} \nu \lambda +y a \,{\mathrm e}^{\lambda \textit {\_a}}\right )}{\mu y \lambda x}}}{\textit {\_a}}d \textit {\_a} +f_{1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{\frac {b x \,{\mathrm e}^{\mu y}}{\mu y}+\frac {y a \,{\mathrm e}^{\lambda x}}{x \lambda }}\]

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6.5.6.7 [1245] Problem 7

problem number 1245

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]];
 

\[\left \{\left \{w(x,y)\to \exp \left (-\frac {(k+1) y^{k+1} \left (\left (y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{k+1},\frac {k+2}{k+1},\frac {a \lambda y^{k+1}}{a \lambda y^{k+1}-b e^{\lambda x} (k+1)}\right )}{a \lambda y^{k+1}-b (k+1) e^{\lambda x}}\right ) \left (\int _1^x\frac {c \exp \left (\frac {(k+1) \left (a \lambda y^{k+1}-b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{k+1},\frac {k+2}{k+1},1-\frac {b e^{\lambda K[1]} (k+1)}{b e^{\lambda x} (k+1)-a \lambda y^{k+1}}\right ) \left (\left (y^{k+1}-\frac {b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a \lambda \left (a \lambda y^{k+1}-b e^{\lambda x} (k+1)\right )}+\beta K[1]\right ) \left (\left (y^{k+1}-\frac {b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b e^{\lambda x}}{a \lambda }\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {\left (f_{1} \left (y^{1+k}-\frac {b \,{\mathrm e}^{\lambda x} k}{\lambda a}-\frac {b \,{\mathrm e}^{\lambda x}}{\lambda a}\right ) a +c \int _{}^{x}{\left (\left (\frac {a \,y^{1+k} \lambda +b \left (1+k \right ) \left ({\mathrm e}^{\lambda \textit {\_a}}-{\mathrm e}^{\lambda x}\right )}{\lambda a}\right )^{\frac {1}{1+k}}\right )}^{-k} {\mathrm e}^{\beta \textit {\_a} -\frac {\int {\left (\left (\frac {a \,y^{1+k} \lambda +b \left (1+k \right ) \left ({\mathrm e}^{\lambda \textit {\_a}}-{\mathrm e}^{\lambda x}\right )}{\lambda a}\right )^{\frac {1}{1+k}}\right )}^{-k}d \textit {\_a}}{a}}d \textit {\_a} \right ) {\mathrm e}^{\frac {\int _{}^{x}{\left (\left (\frac {a \,y^{1+k} \lambda +b \left (1+k \right ) \left ({\mathrm e}^{\lambda \textit {\_a}}-{\mathrm e}^{\lambda x}\right )}{\lambda a}\right )^{\frac {1}{1+k}}\right )}^{-k}d \textit {\_a}}{a}}}{a}\]

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6.5.6.8 [1246] Problem 8

problem number 1246

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]];
 

\[\left \{\left \{w(x,y)\to \frac {e^{-\frac {e^{-\lambda x}}{a \lambda }} \left (-c \operatorname {ExpIntegralEi}\left (\frac {e^{-\lambda x}}{a \lambda }\right )+a \lambda c_1\left (y e^{\frac {b e^{-\lambda x}}{a \lambda }}\right )\right )}{a \lambda }\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {\left (f_{1} \left (y \,{\mathrm e}^{\frac {b \,{\mathrm e}^{-\lambda x}}{\lambda a}}\right ) \lambda a +c \,\operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{-\lambda x}}{\lambda a}\right )\right ) {\mathrm e}^{-\frac {{\mathrm e}^{-\lambda x}}{\lambda a}}}{\lambda a}\]

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6.5.6.9 [1247] Problem 9

problem number 1247

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]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {(k+1) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{k+1},1+\frac {1}{k+1},\frac {b \lambda x^{k+1}}{b \lambda x^{k+1}-a e^{\lambda y} (k+1)}\right )}{a (k+1) e^{\lambda y}-b \lambda x^{k+1}}\right ) \left (\int _1^x\frac {c \exp \left (\left (\beta -\frac {(k+1) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{k+1},1+\frac {1}{k+1},\frac {b \lambda K[1]^{k+1}}{b \lambda x^{k+1}-a e^{\lambda y} (k+1)}\right )}{a e^{\lambda y} (k+1)-b \lambda x^{k+1}}\right ) K[1]\right ) (k+1)}{a e^{\lambda y} (k+1)+b \lambda \left (K[1]^{k+1}-x^{k+1}\right )}dK[1]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b x^{k+1}}{a k+a}\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \left (c \left (k +1\right ) \int _{}^{x}\frac {{\mathrm e}^{\left (-k -1\right ) \int \frac {1}{\left (k +1\right ) a \,{\mathrm e}^{\lambda y}-b \lambda \left (-\textit {\_a}^{k} \textit {\_a} +x^{k} x \right )}d \textit {\_a} +\beta \textit {\_a}}}{\left (k +1\right ) a \,{\mathrm e}^{\lambda y}+\lambda b \left (\textit {\_a}^{k} \textit {\_a} -x^{k} x \right )}d \textit {\_a} +f_{1} \left (\frac {\left (k +1\right ) a \,{\mathrm e}^{\lambda y}-b \lambda x \,x^{k}}{\left (k +1\right ) b \lambda }\right )\right ) {\mathrm e}^{\left (k +1\right ) \int _{}^{x}\frac {1}{\left (k +1\right ) a \,{\mathrm e}^{\lambda y}-b \lambda \left (-\textit {\_a}^{k} \textit {\_a} +x^{k} x \right )}d \textit {\_a}}\]

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6.5.6.10 [1248] Problem 10

problem number 1248

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'));
 

\[w \left (x , y\right ) = \frac {\left ({\mathrm e}^{\beta x}\right )^{\frac {1}{{\mathrm e}^{\lambda y} a \beta -{\mathrm e}^{\beta x} \lambda b}} \left (f_{1} \left (\frac {{\mathrm e}^{\lambda y} a}{\lambda b}-\frac {{\mathrm e}^{\beta x}}{\beta }\right ) \lambda b +\beta c \int _{}^{x}\textit {\_a}^{k} \left (\frac {{\mathrm e}^{\lambda y} a \beta -\lambda b \left ({\mathrm e}^{\beta x}-{\mathrm e}^{\beta \textit {\_a}}\right )}{\lambda b}\right )^{\frac {-{\mathrm e}^{\lambda y} a \beta +{\mathrm e}^{\beta x} \lambda b +1}{{\mathrm e}^{\lambda y} a \beta -{\mathrm e}^{\beta x} \lambda b}} \left ({\mathrm e}^{\beta \textit {\_a}}\right )^{-\frac {1}{{\mathrm e}^{\lambda y} a \beta -{\mathrm e}^{\beta x} \lambda b}}d \textit {\_a} \right ) \left (\frac {{\mathrm e}^{\lambda y} a \beta }{\lambda b}\right )^{-\frac {1}{{\mathrm e}^{\lambda y} a \beta -{\mathrm e}^{\beta x} \lambda b}}}{\lambda b}\]

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