6.2.6 3.1

6.2.6.1 [533] problem number 1

problem number 533

Added January 2, 2019.

Problem 2.3.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + a e^{\lambda x} w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Exp[lambda*x]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+ a*exp(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y\lambda -a{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) \]

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6.2.6.2 [534] problem number 2

problem number 534

Added January 7, 2019.

Problem 2.3.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a e^{\lambda x} +b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Exp[lambda*x] + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*exp(lambda*x)+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-a{{\rm e}^{\lambda \,x}}-\lambda \, \left ( xb-y \right ) }{\lambda }} \right ) \]

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6.2.6.3 [535] problem number 3

problem number 535

Added January 7, 2019.

Problem 2.3.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a e^{\lambda y} +b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Exp[lambda*y] + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*exp(lambda*y)+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\ln \left ( b{{\rm e}^{\lambda \, \left ( xb-y \right ) }}+{{\rm e}^{xb\lambda }}a \right ) }{\lambda \,b}} \right ) \]

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6.2.6.4 [536] problem number 4

problem number 536

Added January 7, 2019.

Problem 2.3.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a e^{\lambda y+ \beta x} +b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Exp[lambda*y + beta*x] + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*exp(lambda*y+beta*x)+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-\ln \left ( \left ( a{{\rm e}^{\beta \,x+y\lambda }}\lambda +\lambda \,b+\beta \right ) ^{-1} \right ) +\lambda \, \left ( xb-y \right ) }{\lambda \,b+\beta }} \right ) \]

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6.2.6.5 [537] problem number 5

problem number 537

Added January 7, 2019.

Problem 2.3.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a e^{\lambda y+ \beta x} +b e^{\gamma x}\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Exp[lambda*y + beta*x] + b*Exp[gamma*x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*exp(lambda*y+beta*x)+b*exp(g*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\lambda } \left ( -a\int \!{{\rm e}^{\beta \,x+{\frac {\lambda \,b{{\rm e}^{gx}}}{g}}}}\,{\rm d}x\lambda -{{\rm e}^{{\frac {\lambda \, \left ( b{{\rm e}^{gx}}-gy \right ) }{g}}}} \right ) } \right ) \]

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6.2.6.6 [538] problem number 6

problem number 538

Added January 7, 2019.

Problem 2.3.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b e^{\beta y} w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[lambda*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := a*exp(lambda*x)*diff(w(x,y),x)+ b*exp(beta*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac { \left ( a\lambda \,{{\rm e}^{\lambda \,x}}-{{\rm e}^{\beta \,y}}b\beta \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}}{b\beta \,\lambda }} \right ) \]

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6.2.6.7 [539] problem number 7

problem number 539

Added January 7, 2019.

Problem 2.3.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a e^{\lambda x} +b \right ) w_x + \left ( c e^{\beta x}+d \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a*Exp[lambda*x] + b)*D[w[x, y], x] + (c + Exp[beta*x] + d)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\beta (c+d) \log \left (a e^{\lambda x}+b\right )-\lambda e^{\beta x} \, _2F_1\left (1,\frac {\beta }{\lambda };\frac {\beta +\lambda }{\lambda };-\frac {a e^{\lambda x}}{b}\right )-\beta \lambda (-b y+c x+d x)}{b \beta \lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
pde := (a*exp(lambda*x)+b)*diff(w(x,y),x)+ (c+exp(beta*x)+d)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {c+{{\rm e}^{\beta \,x}}+d}{a{{\rm e}^{\lambda \,x}}+b}}\,{\rm d}x+y \right ) \]

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6.2.6.8 [540] problem number 8

problem number 540

Added January 7, 2019.

Problem 2.3.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a e^{\lambda x} +b \right ) w_x + \left ( c e^{\beta y}+d \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a*Exp[lambda*x] + b)*D[w[x, y], x] + (c + Exp[beta*y] + d)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := (a*exp(lambda*x)+b)*diff(w(x,y),x)+ (c+exp(beta*y)+d)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\lambda \,b\beta \, \left ( c+d \right ) } \left ( -\RootOf \left ( {{\rm e}^{{\frac {c{\it \_Z}}{c+d}}}} \left ( -{{\rm e}^{{\frac {y\beta \,c}{c+d}}}} \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \,{c}^{2}}{b\lambda \, \left ( c+d \right ) }}} \left ( \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \,cd}{b\lambda \, \left ( c+d \right ) }}} \right ) ^{2}{{\rm e}^{{\frac {y\beta \,d}{c+d}}}} \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \,{d}^{2}}{b\lambda \, \left ( c+d \right ) }}}+ \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \, \left ( c+d \right ) }{\lambda \,b}}}{{\rm e}^{{\frac {c{\it \_Z}}{c+d}}}}{{\rm e}^{{\frac {{\it \_Z}\,d}{c+d}}}}- \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \, \left ( c+d \right ) }{\lambda \,b}}}c- \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \, \left ( c+d \right ) }{\lambda \,b}}}d \right ) \right ) \lambda \,b+ \left ( \left ( c+d \right ) \ln \left ( a{{\rm e}^{\lambda \,x}}+b \right ) +\lambda \, \left ( by-x \left ( c+d \right ) \right ) \right ) \beta \right ) } \right ) \] Has RootOf

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6.2.6.9 [541] problem number 9

problem number 541

Added January 7, 2019.

Problem 2.3.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a e^{\lambda y} +b \right ) w_x + \left ( c e^{\beta x}+d \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a*Exp[lambda*y] + b)*D[w[x, y], x] + (c + Exp[beta*x] + d)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := (a*exp(lambda*y)+b)*diff(w(x,y),x)+ (c+exp(beta*x)+d)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {a{{\rm e}^{y\lambda }}\beta + \left ( -{{\rm e}^{\beta \,x}}+ \left ( \left ( -c-d \right ) x+by \right ) \beta \right ) \lambda }{\beta \,\lambda }} \right ) \]

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6.2.6.10 [542] problem number 10

problem number 542

Added January 7, 2019.

Problem 2.3.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a e^{\lambda x} +b e^{\beta y}\right ) w_x + a \lambda e^{\lambda x} w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*Exp[lambda*x] + b*Exp[beta*y])*D[w[x, y], x] + a*lambda*Exp[lambda*x]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := (a*exp(lambda*x)+b*exp(beta*y))*diff(w(x,y),x)+ a*lambda*exp(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {\lambda \,x+\ln \left ( -b{{\rm e}^{\beta \,y-\lambda \,x}}+a \left ( \beta -1 \right ) \right ) -y}{\beta -1}} \right ) \]

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6.2.6.11 [543] problem number 11

problem number 543

Added January 7, 2019.

Problem 2.3.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a e^{\lambda x+\beta y} +c \mu \right ) w_x - \left ( b e^{\gamma x+ mu y}+c \lambda \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*Exp[lambda*x + beta*y] + c*mu)*D[w[x, y], x] - (b*Exp[gamma*x + mu*y] + c*lambda)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  (a*exp(lambda*x+beta*y)+c*mu)*diff(w(x,y),x)- (b*exp(g*x+ mu*y)+c*lambda)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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