6.4.2 2.1

6.4.2.1 [1023] Problem 1

problem number 1023

Added Feb. 17, 2019.

Problem Chapter 4.2.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c w \]

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x) +b*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'));
 

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

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6.4.2.2 [1024] Problem 2

problem number 1024

Added Feb. 17, 2019.

Problem Chapter 4.2.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + y w_y = b w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.2.3 [1025] Problem 3

problem number 1025

Added Feb. 17, 2019.

Problem Chapter 4.2.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + y w_y = a w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = x^{a} \mathit {\_F1} \left (\frac {y}{x}\right )\]

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6.4.2.4 [1026] Problem 4

problem number 1026

Added Feb. 17, 2019.

Problem Chapter 4.2.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x ( a w_x - b w_y ) = c y w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*(D[w[x, y], x] - b*D[w[x, y], y]) == c*y*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

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

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6.4.2.5 [1027] Problem 5

problem number 1027

Added Feb. 17, 2019.

Problem Chapter 4.2.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + y w_y = a x w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.2.6 [1028] Problem 6

problem number 1028

Added Feb. 17, 2019.

Problem Chapter 4.2.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (x-a) w_x + (y-b) w_y = w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.2.7 [1029] Problem 7

problem number 1029

Added Feb. 17, 2019.

Problem Chapter 4.2.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (y+a x) w_x + (y- a x) w_y = b w \]

Mathematica ✗

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

Failed

Maple ✗

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

time expired

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