111 HFOPDE, chapter 4.5.2

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111.1 Problem 1

problem number 917

Added Feb. 25, 2019.

Problem Chapter 4.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (c x^n+s \ln ^k(\lambda y)) w \]

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 = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*x^n + s*Log[gamma*y]^k)*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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*x^n+s*ln(gamma*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{n}c}{a}}+{\frac {s}{a} \left ( \ln \left ( \gamma \right ) +\ln \left ( {\frac {b{\it \_a}+ya-bx}{a}} \right ) \right ) ^{k}}{d{\it \_a}}}} \]

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111.2 Problem 2

problem number 918

Added Feb. 25, 2019.

Problem Chapter 4.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = (b y^2+c x^n y+ s \ln ^k(\lambda x)) w \]

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 = D[w[x, y], x] + a*D[w[x, y], y] == (b*y^2 + c*x^n*y + s*Log[lambda*x]^k)*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\frac {s \log ^k(\lambda x) (-\log (\lambda x))^{-k} \text {Gamma}(k+1,-\log (\lambda x))}{\lambda }+\frac {1}{3} a^2 b x^3+a b x^2 (y-a x)+b x (y-a x)^2+x^n \left (\frac {a c x^2}{n+2}+\frac {c x (y-a x)}{n+1}\right )\right )\right \}\right \} \]

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 :=  diff(w(x,y),x)+a*diff(w(x,y),y) = (b*y^2+c*x^n*y+ s*ln(lambda*x)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ) {{\rm e}^{\int ^{x}\!b{{\it \_a}}^{2}{a}^{2}+ca{{\it \_a}}^{n+1}+2\, \left ( -ax+y \right ) ab{\it \_a}+{{\it \_a}}^{n} \left ( -ax+y \right ) c+ \left ( -ax+y \right ) ^{2}b+s \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}{d{\it \_a}}}} \]

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111.3 Problem 3

problem number 919

Added March 9, 2019.

Problem Chapter 4.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = b \ln ^k(\lambda x) \ln ^n(\beta y) w \]

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 = D[w[x, y], x] + a*D[w[x, y], y] == b*Log[lambda*x]^k*Log[beta*y]^n*w[x, y]; 
 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';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 :=  diff(w(x,y),x)+a*diff(w(x,y),y) = b*ln(lambda*x)^k*ln(beta*y)^n*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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111.4 Problem 4

problem number 920

Added March 9, 2019.

Problem Chapter 4.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+b x^n) w_y = c \ln ^k(\lambda x) w \]

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 = D[w[x, y], x] + (a*y + b*x^n)*D[w[x, y], y] == c*Log[lambda*x]^k*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{{\rm e}^{-ax}} \left ( \left ( ax \right ) ^{-n/2} \WhittakerM \left ( n/2,n/2+1/2,ax \right ) {x}^{n}{{\rm e}^{1/2\,ax}}b-any-ya \right ) }{a \left ( n+1 \right ) }} \right ) {{\rm e}^{\int \!c \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x}} \]

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111.5 Problem 5

problem number 921

Added March 9, 2019.

Problem Chapter 4.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = x^k (n \ln x+ m \ln y) w \]

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 = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == x^k*(n*Log[x] + m*Log[y])*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right ) \exp \left (\frac {x^k (a k m \log (y)+a k n \log (x)-a n-b m)}{a^2 k^2}\right )\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \left ( {x}^{{\frac {b}{a}}} \right ) ^{{\frac {m{x}^{k}}{ak}}} \left ( y{x}^{-{\frac {b}{a}}} \right ) ^{{\frac {m{x}^{k}}{ak}}}{x}^{{\frac {{x}^{k}n}{ak}}}{{\rm e}^{-1/2\,{\frac {{x}^{k}}{{a}^{2}{k}^{2}} \left ( i\pi \,m \left ( {\it csgn} \left ( iy \right ) \right ) ^{3}ak-i\pi \,m \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}{\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) ak-i\pi \,m{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}ak+i\pi \,m{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) {\it csgn} \left ( iy \right ) {\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) ak+2\,an+2\,bm \right ) }}} \]

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111.6 Problem 6

problem number 922

Added March 9, 2019.

Problem Chapter 4.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^k w_x + b y^n w_y = (c \ln ^m(\lambda x)+s \ln ^t(\beta y)) w \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*x^k*D[w[x, y], x] + b*y^n*D[w[x, y], y] == (c*Log[lambda*x]^m + s*Log[beta*y]^t)*w[x, y]; 
 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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
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*x^k*diff(w(x,y),x)+ b*y^n*diff(w(x,y),y) = (c*ln(lambda*x)^m+s*ln(beta*y)^t)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{x}^{1-k}b \left ( n-1 \right ) +{y}^{-n+1}a \left ( k-1 \right ) }{a \left ( k-1 \right ) }} \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{-k}}{a} \left ( c \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{m}+ \left ( \ln \left ( \beta \, \left ( {\frac {b \left ( n-1 \right ) {{\it \_a}}^{1-k}-{x}^{1-k}b \left ( n-1 \right ) +{y}^{-n+1}a \left ( k-1 \right ) }{a \left ( k-1 \right ) }} \right ) ^{- \left ( n-1 \right ) ^{-1}} \right ) \right ) ^{t}s \right ) }{d{\it \_a}}}} \]