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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {e^{\frac {b x^3}{3}} \left (\left (b x^2+y\right ) \text {HeunT}\left [a^2,-(a-2) b,0,0,b,x\right ]+\text {HeunTPrime}\left [a^2,-(a-2) b,0,0,b,x\right ]\right )}{y \text {HeunT}\left [a^2,-a b,0,0,-b,x\right ]+\text {HeunTPrime}\left [a^2,-a b,0,0,-b,x\right ]}\right )\right \}\right \}\] But it can’t solve it when assuming \(b>0\) which is strange.
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 3\,{ \left ( \left ( {x}^{2}{\it csgn} \left ( b \right ) b-{x}^{2}b-2\,y \right ) {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) -2/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}{\it HeunTPrime} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \left ( -3\, \left ( \left ( {x}^{2}{\it csgn} \left ( b \right ) b-{x}^{2}b-2\,y \right ) {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) -2/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}{\it HeunTPrime} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt {{b}^{2}}}} \left ( {\it HeunT} \left ( -{\frac {{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac { \left ( a-1 \right ) \sqrt {{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}x+6\,{{\rm e}^{1/3\,{x}^{3}\sqrt {{b}^{2}}}} \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt [3]{3} b \text {Gamma}\left (\frac {4}{3},\frac {a x^3}{3}\right )}{a^{4/3}}+\frac {c \text {Gamma}\left (\frac {1}{3},\frac {a x^3}{3}\right )}{3^{2/3} \sqrt [3]{a}}+y e^{-\frac {a x^3}{3}}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -3/4\,{\frac {{{\rm e}^{-1/3\,a{x}^{3}}} \left ( {{\rm e}^{1/6\,a{x}^{3}}}x\sqrt [6]{3} \left ( ca+b \right ) \WhittakerM \left ( 1/6,2/3,1/3\,a{x}^{3} \right ) +4/3\,a\sqrt [6]{a{x}^{3}} \left ( cx-y \right ) \right ) }{a\sqrt [6]{a{x}^{3}}}} \right ) \]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {2 a x^3}{3}}}{y^2}+\frac {i \left (\sqrt {3}+i\right ) b \text {Gamma}\left (\frac {1}{3},-\frac {2 a x^3}{3}\right )}{\sqrt [3]{2} 3^{2/3} \sqrt [3]{a}}\right )\right \}\right \}\]
Maple ✓
restart; 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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{-2\,{b}^{2}+8\,a} \left ( -2\,\sqrt {{b}^{2}-4\,a}b\arctanh \left ( {\frac {\sqrt {{b}^{2}-4\,a} \left ( 2\,axy+b \right ) }{-{b}^{2}+4\,a}} \right ) -8\, \left ( -1/4\,{b}^{2}+a \right ) \left ( \ln \left ( xy \right ) -1/2\,\ln \left ( {x}^{2} \left ( a{x}^{2}{y}^{2}+bxy+1 \right ) \right ) \right ) \right ) } \right ) \]
____________________________________________________________________________________
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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,{\frac {1}{bA}\sum _{{\it \_R}=\RootOf \left ( A{b}^{4}{{\it \_Z}}^{3}+3\,A{b}^{3}c{{\it \_Z}}^{2}+3\,A{b}^{2}{c}^{2}{\it \_Z}+Ab{c}^{3}+a \right ) }{\frac {1}{ \left ( {\it \_R}\,b+c \right ) ^{2}}\ln \left ( {\frac { \left ( y-{\it \_R} \right ) b+ax}{b}} \right ) }}+x \right ) \] 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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{x}^{2}b+{b}^{3}\sum _{{\it \_R}=\RootOf \left ( {c}^{2}a{{\it \_Z}}^{3}+{b}^{3}{\it \_Z}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( {\frac {-bxy-{\it \_R}\,c}{c}} \right ) } \right ) \] 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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^{\sqrt {-4 a c+b^2+2 b+1}} \left (\sqrt {-4 a c+b^2+2 b+1}+2 a x y+b+1\right )}{\sqrt {-4 a c+b^2+2 b+1}-2 a x y-b-1}\right )\right \}\right \}\]
Maple ✓
restart; 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["Global`*"]; 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 ✓
restart; 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["Global`*"]; 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]];
$Aborted
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,b{y}^{4}-1/4\,c{x}^{4}-axy \right ) \]
____________________________________________________________________________________