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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-a \lambda \left (\int {\mathrm e}^{\frac {b \lambda \,{\mathrm e}^{g x}}{g}+\beta x}d x \right )-{\mathrm e}^{\frac {\left (b \,{\mathrm e}^{g x}-g y \right ) \lambda }{g}}}{\lambda }\right )\]

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-b \lambda \RootOf \left (\left (-\left (a \,{\mathrm e}^{\lambda x}+b \right )^{\frac {\beta c^{2}}{\left (c +d \right ) b \lambda }} \left (a \,{\mathrm e}^{\lambda x}+b \right )^{\frac {\beta d^{2}}{\left (c +d \right ) b \lambda }} \left (a \,{\mathrm e}^{\lambda x}+b \right )^{\frac {2 \beta c d}{\left (c +d \right ) b \lambda }} {\mathrm e}^{\frac {\beta c y}{c +d}} {\mathrm e}^{\frac {\beta d y}{c +d}}+\left (a \,{\mathrm e}^{\lambda x}+b \right )^{\frac {\left (c +d \right ) \beta }{b \lambda }} {\mathrm e}^{\frac {\mathit {\_Z} d}{c +d}} {\mathrm e}^{\frac {\mathit {\_Z} c}{c +d}}-c \left (a \,{\mathrm e}^{\lambda x}+b \right )^{\frac {\left (c +d \right ) \beta }{b \lambda }}-d \left (a \,{\mathrm e}^{\lambda x}+b \right )^{\frac {\left (c +d \right ) \beta }{b \lambda }}\right ) {\mathrm e}^{\frac {\mathit {\_Z} c}{c +d}}\right )+\left (\left (b y -\left (c +d \right ) x \right ) \lambda +\left (c +d \right ) \ln \left (a \,{\mathrm e}^{\lambda x}+b \right )\right ) \beta }{\left (c +d \right ) b \beta \lambda }\right )\] 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'));
 

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

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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'));
 

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

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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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