Added Feb. 23, 2019.
Problem Chapter 4.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c e^{\alpha x+ \beta y} w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Exp[alpha*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) e^{\frac {c e^{\alpha x+\beta y}}{a \alpha +b \beta }}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*exp(alpha*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {c{{\rm e}^{\alpha \,x+\beta \,y}}}{a\alpha +b\beta }}}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = (c e^{\lambda x}+ k e^{\mu y}) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*Exp[lambda*x] + k*Exp[mu*y])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) e^{\frac {c e^{\lambda x}}{a \lambda }+\frac {k e^{\mu y}}{b \mu }}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) = (c*exp(lambda*x)+k*exp(mu*y))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {{{\rm e}^{\lambda \,x}}cb\mu +ak\lambda \,{{\rm e}^{\mu \,y}}}{a\lambda \,b\mu }}}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b e^{\beta y} w_y = c w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == c*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {c e^{-\lambda x}}{a \lambda }} c_1\left (\frac {b e^{-\lambda x}}{a \lambda }-\frac {e^{-\beta y}}{\beta }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*x)*diff(w(x,y),x)+b*exp(beta*y)*diff(w(x,y),y) = c*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\lambda \,{{\rm e}^{\lambda \,x}} \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}}{b\beta \,\lambda }} \right ) {{\rm e}^{-{\frac {c{{\rm e}^{-\lambda \,x}}}{a\lambda }}}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda y} w_x + b e^{\beta x} w_y = c w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*y]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == c*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b e^{\beta x}}{a \beta }\right ) \exp \left (\frac {c \left (\beta x-\log \left (\frac {a \beta e^{\lambda y}}{\lambda }\right )\right )}{a \beta e^{\lambda y}-b \lambda e^{\beta x}}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*y)*diff(w(x,y),x)+b*exp(beta*x)*diff(w(x,y),y) = c*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{{\rm e}^{\beta \,x}}b\lambda +{{\rm e}^{y\lambda }}a\beta }{b\beta \,\lambda }} \right ) \left ( {\frac {{{\rm e}^{y\lambda }}a\beta }{\lambda \,b}} \right ) ^{-{\frac {c}{-{{\rm e}^{\beta \,x}}b\lambda +{{\rm e}^{y\lambda }}a\beta }}} \left ( {{\rm e}^{\beta \,x}} \right ) ^{{\frac {c}{-{{\rm e}^{\beta \,x}}b\lambda +{{\rm e}^{y\lambda }}a\beta }}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda x} w_x + b e^{\beta x} w_y = c e^{\gamma y} w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == c*Exp[gamma*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{x (\beta -\lambda )}}{a (\lambda -\beta )}+y\right ) \exp \left (\int _1^x\frac {c \exp \left (y \gamma -\frac {b \left (e^{(\beta -\lambda ) x}-e^{(\beta -\lambda ) K[1]}\right ) \gamma }{a (\beta -\lambda )}-\lambda K[1]\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*x)*diff(w(x,y),x)+b*exp(beta*x)*diff(w(x,y),y) = c*exp(gamma*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-b{{\rm e}^{x \left ( \beta -\lambda \right ) }}+ay \left ( \beta -\lambda \right ) }{ \left ( \beta -\lambda \right ) a}} \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a}{{\rm e}^{1/8\,{\frac {-b{{\rm e}^{x \left ( \beta -\lambda \right ) }}+{{\rm e}^{{\it \_a}\, \left ( \beta -\lambda \right ) }}b+a \left ( \beta -\lambda \right ) \left ( -8\,{\it \_a}\,\lambda +y \right ) }{ \left ( \beta -\lambda \right ) a}}}}}{d{\it \_a}}}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b e^{\beta y} w_y = (c e^{\gamma y} + s e^{\delta y} ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == (c*Exp[gamma*y] + s*Exp[delta*y])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{-\lambda x}}{a \lambda }-\frac {e^{-\beta y}}{\beta }\right ) \exp \left (-\frac {e^{-\lambda x} \left (e^{-\beta y}\right )^{-\frac {\delta +\gamma }{\beta }} \left ((\beta -\delta ) \left (c \gamma \left (e^{-\beta y}\right )^{\frac {\delta }{\beta }} \left (\frac {a \lambda e^{\lambda x-\beta y}}{b \beta }\right )^{\frac {\gamma }{\beta }} \, _2F_1\left (\frac {\beta +\gamma }{\beta },\frac {\gamma }{\beta }-1;\frac {\gamma }{\beta };1-\frac {a e^{\lambda x-\beta y} \lambda }{b \beta }\right )+(\beta -\gamma ) \left (c \left (e^{-\beta y}\right )^{\frac {\delta }{\beta }}+s \left (e^{-\beta y}\right )^{\frac {\gamma }{\beta }}\right )\right )+\delta s (\beta -\gamma ) \left (e^{-\beta y}\right )^{\frac {\gamma }{\beta }} \left (\frac {a \lambda e^{\lambda x-\beta y}}{b \beta }\right )^{\frac {\delta }{\beta }} \, _2F_1\left (\frac {\beta +\delta }{\beta },\frac {\delta }{\beta }-1;\frac {\delta }{\beta };1-\frac {a e^{\lambda x-\beta y} \lambda }{b \beta }\right )\right )}{a \lambda (\beta -\delta ) (\beta -\gamma )}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*x)*diff(w(x,y),x)+b*exp(beta*y)*diff(w(x,y),y) = (c*exp(gamma*y)+s*exp(delta*y))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\lambda \,{{\rm e}^{\lambda \,x}} \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}}{b\beta \,\lambda }} \right ) {{\rm e}^{-8\,{\frac { \left ( a\lambda \,{{\rm e}^{\lambda \,x}}-{{\rm e}^{\beta \,y}}b\beta \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}+{{\rm e}^{-\lambda \,x}}\beta \,b}{b \left ( 8\,\beta -1 \right ) a\lambda \, \left ( \beta -\delta \right ) } \left ( c \left ( \beta -\delta \right ) \left ( {\frac {a\lambda }{ \left ( a\lambda \,{{\rm e}^{\lambda \,x}}-{{\rm e}^{\beta \,y}}b\beta \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}+{{\rm e}^{-\lambda \,x}}\beta \,b}} \right ) ^{1/8\,{\beta }^{-1}}+ \left ( {\frac {a\lambda }{ \left ( a\lambda \,{{\rm e}^{\lambda \,x}}-{{\rm e}^{\beta \,y}}b\beta \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}+{{\rm e}^{-\lambda \,x}}\beta \,b}} \right ) ^{{\frac {\delta }{\beta }}}s \left ( -1/8+\beta \right ) \right ) }}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a e^{\beta x} w_x + (b e^{\gamma x}+c e^{\lambda y} ) w_y = (s e^{\mu x} + k e^{\delta y} + p ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*Exp[beta*x]*D[w[x, y], x] + (b*Exp[gamma*x] + c*Exp[lambda*y])*D[w[x, y], y] == (s*Exp[mu*x] + k*Exp[delta*y] + p)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := a*exp(beta*x)*diff(w(x,y),x)+(b*exp(gamma*x)+c*exp(lambda*y))*diff(w(x,y),y) = (s*exp(mu*x) + k*exp(delta*y) + p)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( \left ( {a \left ( a\int \!{\frac {c}{a}{{\rm e}^{{\frac {-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x\lambda -\int \!{{\rm e}^{{\frac {-8\,b\lambda \,{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}-8\,\beta \,{\it \_b}\, \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}\,{\rm d}{\it \_b}c\lambda +a{{\rm e}^{-8\,{\frac { \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) ^{-1}} \right ) ^{{\frac {\delta }{\lambda }}}k{{\rm e}^{{\frac {-8\,\delta \,b{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}-8\,\beta \,{\it \_b}\, \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}+s{{\rm e}^{{\it \_b}\, \left ( -\beta +\mu \right ) }}+{{\rm e}^{-\beta \,{\it \_b}}}p \right ) }{d{\it \_b}}}}{\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x-{{\rm e}^{-8\,{\frac { \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) } \right ) \]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a e^{\beta x} w_x + (b e^{\gamma x}+c e^{\lambda y} ) w_y = (s e^{\mu x+\delta y} + k ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*Exp[beta*x]*D[w[x, y], x] + (b*Exp[gamma*x] + c*Exp[lambda*y])*D[w[x, y], y] == (s*Exp[mu*x + delta*y] + k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := a*exp(beta*x)*diff(w(x,y),x)+(b*exp(gamma*x)+c*exp(lambda*y))*diff(w(x,y),y) = (s*exp(mu*x+delta*y) + k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x-{{\rm e}^{-8\,{\frac { \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( \left ( {a \left ( a\int \!{\frac {c}{a}{{\rm e}^{{\frac {-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x\lambda -\int \!{{\rm e}^{{\frac {-8\,b\lambda \,{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}-8\,\beta \,{\it \_b}\, \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}\,{\rm d}{\it \_b}c\lambda +a{{\rm e}^{-8\,{\frac { \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) ^{-1}} \right ) ^{{\frac {\delta }{\lambda }}}s{{\rm e}^{{\frac {-8\,\delta \,b{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}+8\,{\it \_b}\, \left ( -1/8+\beta \right ) \left ( -\beta +\mu \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}+{{\rm e}^{-\beta \,{\it \_b}}}k \right ) }{d{\it \_b}}}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a e^{\beta x} w_x + b e^{\gamma x+\lambda y} w_y = (c e^{\mu x+\delta y} + k ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[beta*x]*D[w[x, y], x] + (b*Exp[gamma*x + lambda*y])*D[w[x, y], y] == (c*Exp[mu*x + delta*y] + k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{\gamma x-\beta x}}{a \beta -a \gamma }-\frac {e^{-\lambda y}}{\lambda }\right ) \exp \left (\frac {c (\gamma -\beta ) \left (e^{\lambda y}\right )^{\delta /\lambda } e^{-\gamma x-\lambda y+\mu x} \, _2F_1\left (1,\frac {\mu -\gamma }{\beta -\gamma };\frac {\beta \delta -\gamma \delta -\gamma \lambda +\lambda \mu }{\beta \lambda -\gamma \lambda };1-\frac {a e^{\beta x-\gamma x-\lambda y} (\beta -\gamma )}{b \lambda }\right )}{b (\beta (\lambda -\delta )+\delta \gamma -\lambda \mu )}-\frac {k e^{-\beta x}}{a \beta }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(beta*x)*diff(w(x,y),x)+(b*exp(gamma*x+lambda*y))*diff(w(x,y),y) = (c*exp(mu*x+delta*y) + k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -8\,{\frac { \left ( \left ( -1/8+\beta \right ) a{{\rm e}^{\beta \,x-x/8}}-{{\rm e}^{y\lambda }}\lambda \,b \right ) {{\rm e}^{-\beta \,x-y\lambda +x/8}}}{b\lambda \, \left ( 8\,\beta -1 \right ) }} \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\rm e}^{-\beta \,{\it \_a}}}}{a} \left ( c \left ( {\frac {a \left ( 8\,\beta -1 \right ) ^{2}}{ \left ( 64\,\beta -8 \right ) \left ( \left ( \left ( -1/8+\beta \right ) a{{\rm e}^{\beta \,x-x/8}}-{{\rm e}^{y\lambda }}\lambda \,b \right ) {{\rm e}^{-\beta \,x-y\lambda +x/8}}+b\lambda \,{{\rm e}^{-\beta \,{\it \_a}+{\it \_a}/8}} \right ) }} \right ) ^{{\frac {\delta }{\lambda }}}{{\rm e}^{\mu \,{\it \_a}}}+k \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.3.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a e^{\lambda y} w_x + b e^{\beta x} w_y = (c e^{\mu x} + k ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == (c*Exp[mu*x] + k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {\frac {c e^{x (\mu -\lambda )}}{\lambda -\mu }+\frac {k e^{-\lambda x}}{\lambda }}{a}} c_1\left (\frac {b e^{x (\beta -\lambda )}}{a (\lambda -\beta )}+y\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*x)*diff(w(x,y),x)+b*exp(beta*x)*diff(w(x,y),y) = (c*exp(mu*x) + k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-b{{\rm e}^{x \left ( \beta -\lambda \right ) }}+ay \left ( \beta -\lambda \right ) }{ \left ( \beta -\lambda \right ) a}} \right ) {{\rm e}^{{\frac {c{{\rm e}^{ \left ( -\lambda +\mu \right ) x}}\lambda -{{\rm e}^{-\lambda \,x}}k \left ( -\lambda +\mu \right ) }{a\lambda \, \left ( -\lambda +\mu \right ) }}}}\]
____________________________________________________________________________________