99 HFOPDE, chapter 4.2.1

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99.1 Problem 1

problem number 832

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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99.2 Problem 2

problem number 833

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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99.3 Problem 3

problem number 834

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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99.4 Problem 4

problem number 835

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( bx+y \right ) {x}^{ \left ( bx+y \right ) c}{{\rm e}^{-bcx}} \]

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99.5 Problem 5

problem number 836

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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99.6 Problem 6

problem number 837

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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99.7 Problem 7

problem number 838

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

\[ \text { sol=() } \]