50 HFOPDE, chapter 2.5.1

____________________________________________________________________________________

50.1 problem number 1

problem number 422

Added January 14, 2019.

Problem 2.5.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \ln ^k(\lambda x)+ b\right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (a*Log[lambda*x]^k + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {(-\log (\lambda x))^{-k} \left (a \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))+b \lambda x (-\log (\lambda x))^k-\lambda y (-\log (\lambda x))^k\right )}{\lambda }\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(a*ln(lambda*x)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+y \right ) \]

____________________________________________________________________________________

50.2 problem number 3

problem number 423

Added January 14, 2019.

Problem 2.5.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \ln ^k(\lambda y)+ b\right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (a*Log[lambda*y]^k + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \frac {1}{a \log ^k(\lambda K[1])+b} \, dK[1]-x\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(a*ln(lambda*y)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \ln \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]

____________________________________________________________________________________

50.3 problem number 4

problem number 424

Added January 14, 2019.

Problem 2.5.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \ln ^k(x+\lambda y)\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + a*Log[x + lambda*y]^k*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+a*ln(x+lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {y\lambda +x}{\lambda }}}\! \left ( 1+a \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}\lambda +x \right ) \]