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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y \lambda -a \,{\mathrm e}^{\lambda x}}{\lambda }\right )\]

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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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to \frac {c_1 \left (\beta \log \left (a \lambda e^{\beta x+\lambda y}+b \lambda +\beta \right )+b \lambda \log \left (\beta (b \lambda +\beta ) e^{\beta x+\lambda y}\right )\right )}{\beta (b \lambda +\beta )}-\frac {c_1 \lambda y}{\beta }+c_2\right \}\right \}\]

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'));
 

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

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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'));
 

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

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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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\beta \left ((c+d) \log \left (a e^{\lambda x}+b\right )+\log (b) (c+d)+b \lambda y-c \lambda x+c \log (\lambda )-d \lambda x+d \log (\lambda )\right )-\lambda e^{\beta x} \operatorname {Hypergeometric2F1}\left (1,\frac {\beta }{\lambda },\frac {\beta +\lambda }{\lambda },-\frac {a e^{\lambda x}}{b}\right )}{b \beta \lambda }\right )\right \}\right \}\]

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'));
 

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

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\lambda b \operatorname {RootOf}\left (\left (a \,{\mathrm e}^{\lambda x}+b \right )^{\frac {\beta d}{b \lambda }} {\mathrm e}^{\frac {-d x \beta c -x \beta \,d^{2}+\textit {\_Z} b c}{b \left (c +d \right )}} \left ({\mathrm e}^{-\frac {y \beta c}{c +d}} c +{\mathrm e}^{-\frac {y \beta c}{c +d}} d -{\mathrm e}^{\frac {-y \beta c +c \textit {\_Z} +d \textit {\_Z}}{c +d}}+{\mathrm e}^{\frac {d y \beta }{c +d}}\right )\right )+\left (\left (c +d \right ) \ln \left (a \,{\mathrm e}^{\lambda x}+b \right )+\left (b y -x \left (c +d \right )\right ) \lambda \right ) \beta }{\lambda b \beta \left (c +d \right )}\right )\]

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'));
 

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

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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'));
 

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

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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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