____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(y^2+b x^2 y-a^2-a b x^2)w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = D[w[x, y], x] + (y^2 + b*x^2*y - a^2 - a*b*x)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {b x^3+2 x y+2}{b x^2+2 y}\right )\right \}\right \} \] But it can’t solve it when assuming \(b>0\) which is strange.
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := diff(w(x,y),x)+ (y^2+b*x^2*y-a^2-a*b*x)*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 ( {1 \left ( -3\,{\it csgn} \left ( b \right ) {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) b{x}^{2}+2\,{3}^{2/3}\sqrt [6]{{b}^{2}}{\it HeunTPrime} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) +3\,{\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) b{x}^{2}+6\,{\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) y \right ) {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \left ( 3\,{\it csgn} \left ( b \right ) \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{2}\int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt {{b}^{2}}}} \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}xb{x}^{2}-2\,{3}^{2/3}{\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) {\it HeunTPrime} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt {{b}^{2}}}} \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}x\sqrt [6]{{b}^{2}}-3\, \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{2}\int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt {{b}^{2}}}} \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}xb{x}^{2}-6\, \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{2}\int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt {{b}^{2}}}} \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}xy-6\,{{\rm e}^{1/3\,{x}^{3}\sqrt {{b}^{2}}}} \right ) ^{-1}} \right ) \] Mathematica solution is much simpler
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a x^2 y+b x^3+c) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = D[w[x, y], x] + (a*x^2*y + b*x^3 + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {a x^3}{3}} \left (\sqrt [3]{3} b e^{\frac {a x^3}{3}} \text {Gamma}\left (\frac {1}{3},\frac {a x^3}{3}\right )+\sqrt [3]{3} a c e^{\frac {a x^3}{3}} \text {Gamma}\left (\frac {1}{3},\frac {a x^3}{3}\right )+3 a^{4/3} y+3 \sqrt [3]{a} b x\right )}{3 a^{4/3}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde := diff(w(x,y),x)+ (a*x^2*y+b*x^3+c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,{\frac { \left ( 3\,\sqrt [6]{3} \WhittakerM \left ( 1/6,2/3,1/3\,a{x}^{3} \right ) {{\rm e}^{1/6\,a{x}^{3}}}acx+3\,\sqrt [6]{3} \WhittakerM \left ( 1/6,2/3,1/3\,a{x}^{3} \right ) {{\rm e}^{1/6\,a{x}^{3}}}bx+4\,acx\sqrt [6]{a{x}^{3}}-4\,ya\sqrt [6]{a{x}^{3}} \right ) {{\rm e}^{-1/3\,a{x}^{3}}}}{a\sqrt [6]{a{x}^{3}}}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a x^2 y+b y^3) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = D[w[x, y], x] + (a*x^2*y + b*y^3)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {(-1)^{2/3} \left (-2^{2/3} \sqrt [3]{3} b y^2 \text {Gamma}\left (\frac {1}{3},-\frac {2 a x^3}{3}\right )+3 \sqrt [3]{a} e^{\frac {2 a x^3}{3}+\frac {i \pi }{3}}\right )}{3 \sqrt [3]{a} y^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := diff(w(x,y),x)+ (a*x^2*y+b*y^3)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/9\,{\frac {2\,{2}^{2/3}{3}^{5/6}bxO{y}^{2}-3\,{2}^{2/3}\sqrt [3]{3}bx\Gamma \left ( 1/3,-2/3\,a{x}^{3} \right ) \Gamma \left ( 2/3 \right ) {y}^{2}+9\,O\Gamma \left ( 2/3 \right ) {{\rm e}^{2/3\,a{x}^{3}}}}{O\Gamma \left ( 2/3 \right ) {y}^{2}}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a x y+b) y^2 w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a, b]; pde = D[w[x, y], x] + (a*x*y + b)*y^2*D[w[x, y], y] == 0; 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'; pde := diff(w(x,y),x)+ (a*x*y+b)*y^2*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {1}{-{b}^{2}+4\,a} \left ( 2\,\sqrt {{b}^{2}-4\,a}b\arctanh \left ( {\frac {\sqrt {{b}^{2}-4\,a} \left ( 2\,axy+b \right ) }{-{b}^{2}+4\,a}} \right ) +\ln \left ( {x}^{2} \left ( a{x}^{2}{y}^{2}+bxy+1 \right ) \right ) {b}^{2}-2\,\ln \left ( yx \right ) {b}^{2}-4\,\ln \left ( {x}^{2} \left ( a{x}^{2}{y}^{2}+bxy+1 \right ) \right ) a+8\,\ln \left ( yx \right ) a \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +A(a x+b y+c)^3 y^2 w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a, b, A]; pde = D[w[x, y], x] + A*(a*x + b*y + c)^3*D[w[x, y], y] == 0; 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';A:='A'; pde := diff(w(x,y),x)+ A*(a*x+b*y+c)^3*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,{\frac {1}{Ab}\sum _{{\it \_R}=\RootOf \left ( A{b}^{4}{{\it \_Z}}^{3}+3\,A{b}^{3}c{{\it \_Z}}^{2}+3\,A{b}^{2}{c}^{2}{\it \_Z}+Ab{c}^{3}+a \right ) }{\frac {1}{{b}^{2}{{\it \_R}}^{2}+2\,bc{\it \_R}+{c}^{2}}\ln \left ( {\frac {-{\it \_R}\,b+ax+by}{b}} \right ) }}+x \right ) \] Answer contains RootOf
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x +(a x^4 y^3+(b x^2-1)y+c x) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a, b, c]; pde = x*D[w[x, y], x] + (a*x^4*y^3 + (b*x^2 - 1)*y + c*x)*D[w[x, y], y] == 0; 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';c:='c'; pde := x*diff(w(x,y),x)+ (a*x^4*y^3+(b*x^2-1)*y+c*x)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,b{x}^{2}+{b}^{3}\sum _{{\it \_R}=\RootOf \left ( {c}^{2}a{{\it \_Z}}^{3}+{b}^{3}{\it \_Z}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac {bxy+{\it \_R}\,c}{c}} \right ) } \right ) \] Answer contains RootOf
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x^2 w_x +(a x^2 y^2+b x y+c) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = x^2*D[w[x, y], x] + (a*x^2*y^2 + b*x*y + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 a y x^{\sqrt {-4 a c+b^2+2 b+1}+1}+b x^{\sqrt {-4 a c+b^2+2 b+1}}+\sqrt {-4 a c+b^2+2 b+1} x^{\sqrt {-4 a c+b^2+2 b+1}}+x^{\sqrt {-4 a c+b^2+2 b+1}}}{\sqrt {-4 a c+b^2+2 b+1}-2 a x y-b-1}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde := x^2*diff(w(x,y),x)+ (a*x^2*y^2+b*x*y+c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {4\,ca-{b}^{2}-2\,b-1}} \left ( \ln \left ( x \right ) \sqrt {4\,ca-{b}^{2}-2\,b-1}-2\,\arctan \left ( {\frac {2\,axy+b+1}{\sqrt {4\,ca-{b}^{2}-2\,b-1}}} \right ) \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^2 y+b) w_x -(a x y^2+c) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = (a*x^2*y + b)*D[w[x, y], x] - (a*x*y^2 + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a x^2 y^2+2 b y+2 c x}{a}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde := (a*x^2*y+b)*diff(w(x,y),x)- (a*x*y^2+c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,a{x}^{2}{y}^{2}-by-cx \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b y^3) w_x -(c x^3+a y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a, b, c]; pde = (a*x + b*y^3)*D[w[x, y], x] - (c*x^3 + a*y)*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';c:='c'; pde := (a*x+b*y^3)*diff(w(x,y),x)- (c*x^3+a*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,b{y}^{4}-1/4\,{x}^{4}c-axy \right ) \]