100 HFOPDE, chapter 4.2.2

____________________________________________________________________________________

100.1 Problem 1

problem number 839

Added Feb. 17, 2019.

Problem Chapter 4.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (x^2-y^2) w \]

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] == (x^2 - y^2)*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y-b x}{a}\right ) \exp \left (-\frac {b^2 x^3}{3 a^3}-\frac {b x^2 (a y-b x)}{a^3}-\frac {x (a y-b x)^2}{a^3}+\frac {x^3}{3 a}\right )\right \}\right \} \]

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) = (x^2-y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{-{\frac { \left ( ya-bx \right ) ^{2}x}{{a}^{3}}}-{\frac { \left ( ya-bx \right ) b{x}^{2}}{{a}^{3}}}+1/3\,{\frac {{x}^{3}}{a}}-1/3\,{\frac {{x}^{3}{b}^{2}}{{a}^{3}}}}} \]

____________________________________________________________________________________

100.2 Problem 2

problem number 840

Added Feb. 17, 2019.

Problem Chapter 4.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + a x y w_y = b y^2 w \]

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^2*D[w[x, y], x] + a*x*y*D[w[x, y], y] == b*y^2*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {b y^2}{(2 a-1) x}} c_1\left (y x^{-a}\right )\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-a} \right ) {{\rm e}^{{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}} \]

____________________________________________________________________________________

100.3 Problem 3

problem number 841

Added Feb. 17, 2019.

Problem Chapter 4.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^2 w_x + b y^2 w_y = (x+c y) w \]

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

\[ \left \{\left \{w(x,y)\to x^{\frac {1}{a}+\frac {c}{b}} \left (b-\frac {b y-a x}{y}\right )^{-\frac {c}{b}} c_1\left (\frac {b y-a x}{a x y}\right )\right \}\right \} \]

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

\[ w \left ( x,y \right ) = \left ( -{\frac {-ax+by}{y}}+b \right ) ^{-{\frac {c}{b}}}{x}^{{\frac {c}{b}}+{a}^{-1}}{\it \_F1} \left ( -{\frac {-ax+by}{yax}} \right ) \]

____________________________________________________________________________________

100.4 Problem 4

problem number 842

Added Feb. 17, 2019.

Problem Chapter 4.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + a y^2 w_y = (b x^2+c x y+d y^2) w \]

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

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {ya-x}{yx}} \right ) \left ( -{\frac {ya-x}{y}}+a \right ) ^{-{\frac {cxy}{ya-x}}}{{\rm e}^{-{\frac {yx}{ya-x} \left ( {\frac { \left ( ya-x \right ) ^{2}b}{{y}^{2}}}-{\frac { \left ( ya-x \right ) ab}{y}}-d \right ) \left ( -{\frac {ya-x}{y}}+a \right ) ^{-1}}}} \]

____________________________________________________________________________________

100.5 Problem 5

problem number 843

Added Feb. 17, 2019.

Problem Chapter 4.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ y^2 w_x + a x^2 w_y = (b x^2+c y^2) w \]

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

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{3} \left (y^3-a x^3\right )\right ) e^{\frac {b \sqrt [3]{y^3}}{a}+c x}\right \},\left \{w(x,y)\to c_1\left (\frac {1}{3} \left (y^3-a x^3\right )\right ) e^{c x-\frac {\sqrt [3]{-1} b \sqrt [3]{y^3}}{a}}\right \},\left \{w(x,y)\to c_1\left (\frac {1}{3} \left (y^3-a x^3\right )\right ) e^{\frac {(-1)^{2/3} b \sqrt [3]{y^3}}{a}+c x}\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -a{x}^{3}+{y}^{3} \right ) {{\rm e}^{{\frac {cax+by}{a}}}} \]

____________________________________________________________________________________

100.6 Problem 6

problem number 844

Added Feb. 17, 2019.

Problem Chapter 4.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x y w_x + a y^2 w_y = (b x+c y + d) w \]

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

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-a} \right ) {x}^{c}{{\rm e}^{-{\frac {bx}{ \left ( a-1 \right ) y}}-{\frac {d}{ \left ( a-1 \right ) y}}+{\frac {d}{ \left ( a-1 \right ) ya}}}} \]

____________________________________________________________________________________

100.7 Problem 7

problem number 845

Added Feb. 17, 2019.

Problem Chapter 4.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x(a y+b) w_x + (a y^2-b x) w_y = a y w \]

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*y + b)*D[w[x, y], x] + (a*y^2 - b*x)*D[w[x, y], y] == a*y*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \]

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*y+b)*diff(w(x,y),x) +(a*y^2-b*x)*diff(w(x,y),y) =a*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 ) ={{\rm e}^{1/9\,\int ^{x}\!{\frac {1}{{\it \_a}\, \left ( {\it \_a}\,a-b \right ) } \left ( 2\,{{\rm e}^{\RootOf \left ( -2\,\ln \left ( -9/2\,{\frac {ax-b}{ya+b}} \right ) {{\rm e}^{{\it \_Z}}}ax-2\,\ln \left ( -9/2\,{\frac {ax-b}{ya+b}} \right ) {{\rm e}^{{\it \_Z}}}ay-2\,\ln \left ( {\frac { \left ( {\it \_a}\,a-b \right ) \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) }{{\it \_a}}} \right ) {{\rm e}^{{\it \_Z}}}ax-2\,\ln \left ( {\frac { \left ( {\it \_a}\,a-b \right ) \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) }{{\it \_a}}} \right ) {{\rm e}^{{\it \_Z}}}ay+2\,\ln \left ( -9\,{\frac {a \left ( x+y \right ) \left ( ax-b \right ) }{x \left ( ya+b \right ) }} \right ) {{\rm e}^{{\it \_Z}}}ax+2\,\ln \left ( -9\,{\frac {a \left ( x+y \right ) \left ( ax-b \right ) }{x \left ( ya+b \right ) }} \right ) {{\rm e}^{{\it \_Z}}}ay+2\,{\it \_Z}\,{{\rm e}^{{\it \_Z}}}ax+2\,{\it \_Z}\,{{\rm e}^{{\it \_Z}}}ay+9\,\ln \left ( -9/2\,{\frac {ax-b}{ya+b}} \right ) ax+9\,\ln \left ( -9/2\,{\frac {ax-b}{ya+b}} \right ) ay+9\,\ln \left ( {\frac { \left ( {\it \_a}\,a-b \right ) \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) }{{\it \_a}}} \right ) ax+9\,\ln \left ( {\frac { \left ( {\it \_a}\,a-b \right ) \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) }{{\it \_a}}} \right ) ay-9\,\ln \left ( -9\,{\frac {a \left ( x+y \right ) \left ( ax-b \right ) }{x \left ( ya+b \right ) }} \right ) ax-9\,\ln \left ( -9\,{\frac {a \left ( x+y \right ) \left ( ax-b \right ) }{x \left ( ya+b \right ) }} \right ) ya-2\,y{{\rm e}^{{\it \_Z}}}a-9\,ax{\it \_Z}-9\,ya{\it \_Z}-2\,b{{\rm e}^{{\it \_Z}}}-9\,ax+9\,b \right ) }}b+9\,{\it \_a}\,a-9\,b \right ) }{d{\it \_a}}}}{\it \_F1} \left ( -1/3\,{\frac {1}{a \left ( x+y \right ) } \left ( \ln \left ( -9\,{\frac {a \left ( x+y \right ) \left ( ax-b \right ) }{x \left ( ya+b \right ) }} \right ) ya+\ln \left ( -9\,{\frac {a \left ( x+y \right ) \left ( ax-b \right ) }{x \left ( ya+b \right ) }} \right ) ax-\ln \left ( -9/2\,{\frac {ax-b}{ya+b}} \right ) ay-\ln \left ( -9/2\,{\frac {ax-b}{ya+b}} \right ) ax-ya-b \right ) } \right ) \]

____________________________________________________________________________________

100.8 Problem 8

problem number 846

Added Feb. 17, 2019.

Problem Chapter 4.2.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*(k*y - x + a)*D[w[x, y], x] - y*(k*x - y + a)*D[w[x, y], y] == b*(y - x)*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 :=x*(k*y-x+a)*diff(w(x,y),x)-y*(k*x-y+a)*diff(w(x,y),y) = b*(y-x)*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 ( -1/3\,{\frac {{k}^{2}+k+1}{ \left ( k+1 \right ) k} \left ( \left ( k+1 \right ) \ln \left ( -{\frac { \left ( {k}^{2}+k+1 \right ) \left ( a-x-y \right ) k}{ \left ( k-1 \right ) \left ( ky+a-x \right ) }} \right ) +k\ln \left ( x-a \right ) -k\ln \left ( -{\frac { \left ( k+1 \right ) \left ( {k}^{2}+k+1 \right ) \left ( a-x \right ) }{ \left ( k+2 \right ) \left ( ky+a-x \right ) }} \right ) -\ln \left ( x \right ) -\ln \left ( {\frac {ky \left ( k+1 \right ) \left ( {k}^{2}+k+1 \right ) }{ \left ( 2\,k+1 \right ) \left ( ky+a-x \right ) }} \right ) \right ) } \right ) {{\rm e}^{1/9\,\int ^{x}\!2\,{\frac {b}{ \left ( k+1 \right ) \left ( {k}^{2}+k+1 \right ) \left ( a-{\it \_a} \right ) k{\it \_a}} \left ( \left ( k+1/2 \right ) \left ( {\it \_a}\,k-{\it \_a}+a \right ) \left ( k+2 \right ) \left ( k-1 \right ) \RootOf \left ( {k}^{3}\ln \left ( {\it \_a}-a \right ) -{k}^{3}\ln \left ( x-a \right ) +{k}^{3}\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) -\ln \left ( 2\,{\it \_Z}\,{k}^{2}-{\it \_Z}\,k-3\,{k}^{2}-{\it \_Z}-3\,k-3 \right ) {k}^{3}+{k}^{2}\ln \left ( {\it \_a}-a \right ) -{k}^{2}\ln \left ( x-a \right ) +81\,\int ^{-3\,{\frac { \left ( -{k}^{2}y+2\,ak-2\,kx-2\,ky+a-x \right ) \left ( {k}^{2}+k+1 \right ) }{ \left ( k-1 \right ) \left ( 2\,k+1 \right ) \left ( k+2 \right ) \left ( ky+a-x \right ) }}}\!{\frac { \left ( {k}^{2}+k+1 \right ) ^{3}}{ \left ( 2\,{\it \_a}\,{k}^{2}-{\it \_a}\,k-3\,{k}^{2}-{\it \_a}-3\,k-3 \right ) \left ( {\it \_a}\,{k}^{2}+{\it \_a}\,k+3\,{k}^{2}-2\,{\it \_a}+3\,k+3 \right ) \left ( 2\,{\it \_a}\,{k}^{2}+5\,{\it \_a}\,k-3\,{k}^{2}+2\,{\it \_a}-3\,k-3 \right ) }}{d{\it \_a}}{k}^{2}-{k}^{2}\ln \left ( {\it \_a} \right ) +{k}^{2}\ln \left ( x \right ) +2\,{k}^{2}\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) -{k}^{2}\ln \left ( {\it \_Z}\,{k}^{2}+{\it \_Z}\,k+3\,{k}^{2}-2\,{\it \_Z}+3\,k+3 \right ) -\ln \left ( 2\,{\it \_Z}\,{k}^{2}-{\it \_Z}\,k-3\,{k}^{2}-{\it \_Z}-3\,k-3 \right ) {k}^{2}+k\ln \left ( {\it \_a}-a \right ) -k\ln \left ( x-a \right ) +81\,\int ^{-3\,{\frac { \left ( -{k}^{2}y+2\,ak-2\,kx-2\,ky+a-x \right ) \left ( {k}^{2}+k+1 \right ) }{ \left ( k-1 \right ) \left ( 2\,k+1 \right ) \left ( k+2 \right ) \left ( ky+a-x \right ) }}}\!{\frac { \left ( {k}^{2}+k+1 \right ) ^{3}}{ \left ( 2\,{\it \_a}\,{k}^{2}-{\it \_a}\,k-3\,{k}^{2}-{\it \_a}-3\,k-3 \right ) \left ( {\it \_a}\,{k}^{2}+{\it \_a}\,k+3\,{k}^{2}-2\,{\it \_a}+3\,k+3 \right ) \left ( 2\,{\it \_a}\,{k}^{2}+5\,{\it \_a}\,k-3\,{k}^{2}+2\,{\it \_a}-3\,k-3 \right ) }}{d{\it \_a}}k-k\ln \left ( {\it \_a} \right ) +k\ln \left ( x \right ) +2\,k\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) -k\ln \left ( {\it \_Z}\,{k}^{2}+{\it \_Z}\,k+3\,{k}^{2}-2\,{\it \_Z}+3\,k+3 \right ) -\ln \left ( 2\,{\it \_Z}\,{k}^{2}-{\it \_Z}\,k-3\,{k}^{2}-{\it \_Z}-3\,k-3 \right ) k-\ln \left ( {\it \_a} \right ) +\ln \left ( x \right ) +\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) -\ln \left ( {\it \_Z}\,{k}^{2}+{\it \_Z}\,k+3\,{k}^{2}-2\,{\it \_Z}+3\,k+3 \right ) \right ) +3\, \left ( {k}^{2}+k+1 \right ) \left ( -1/2\,{\it \_a}\,{k}^{2}+ \left ( a-2\,{\it \_a} \right ) k+a/2-{\it \_a}/2 \right ) \right ) }{d{\it \_a}}}} \]