____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a x^2+b x+c) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = D[w[x, y], x] + (a*x^2 + b*x + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{6} \left (-2 a x^3-3 b x^2-6 c x+6 y\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde:=diff(w(x,y),x)+(a*x^2+b*x+c)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{x}^{3}-1/2\,b{x}^{2}-cx+y \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y^2+b y+c) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = D[w[x, y], x] + (a*y^2 + b*y + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {2 a y \sqrt {4 a c-b^2}+b \sqrt {4 a c-b^2}}{4 a c-b^2}\right )-4 a c x+b^2 x}{4 a c-b^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde:=diff(w(x,y),x)+(a*y^2+b*y+c)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {4\,ca-{b}^{2}}} \left ( -x\sqrt {4\,ca-{b}^{2}}+2\,\arctan \left ( {\frac {2\,ya+b}{\sqrt {4\,ca-{b}^{2}}}} \right ) \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y+b x^2+c x) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = D[w[x, y], x] + (a*y + b*x^2 + c*x)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-a x} \left (a^3 y+a^2 b x^2+a^2 c x+2 a b x+a c+2 b\right )}{a^3}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde:=diff(w(x,y),x)+(a*y+b*x^2+c*x)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( b{x}^{2}{a}^{2}+y{a}^{3}+{a}^{2}cx+2\,axb+ca+2\,b \right ) {{\rm e}^{-ax}}}{{a}^{3}}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a x y+b x^2+ c x +k y +s) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c, k, s]; pde = D[w[x, y], x] + (a*x*y + b*x^2 + c*x + k*y + s)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {a x^2}{2}-k x} \left (2 a^{3/2} b x+2 a^{3/2} c+2 a^{5/2} y-\sqrt {2 \pi } a^2 s e^{\frac {(a x+k)^2}{2 a}} \text {Erf}\left (\frac {k}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} x}{\sqrt {2}}\right )-\sqrt {2 \pi } b k^2 e^{\frac {(a x+k)^2}{2 a}} \text {Erf}\left (\frac {k}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} x}{\sqrt {2}}\right )-\sqrt {2 \pi } a b e^{\frac {(a x+k)^2}{2 a}} \text {Erf}\left (\frac {k}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} x}{\sqrt {2}}\right )-2 \sqrt {a} b k+\sqrt {2 \pi } a c k e^{\frac {(a x+k)^2}{2 a}} \text {Erf}\left (\frac {k}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} x}{\sqrt {2}}\right )\right )}{2 a^{5/2}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c';k:='k';s:='s'; pde:=diff(w(x,y),x)+(a*x*y+b*x^2+c*x+k*y+s)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {{{\rm e}^{-x \left ( ax+2\,k \right ) }}}{{a}^{5/2}} \left ( \sqrt {2}{{\rm e}^{1/2\,{\frac {2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac {\sqrt {2} \left ( ax+k \right ) }{\sqrt {a}}} \right ) {a}^{2}s\sqrt {\pi }-\sqrt {2}{{\rm e}^{1/2\,{\frac {2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac {\sqrt {2} \left ( ax+k \right ) }{\sqrt {a}}} \right ) ack\sqrt {\pi }+\sqrt {2}{{\rm e}^{1/2\,{\frac {2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac {\sqrt {2} \left ( ax+k \right ) }{\sqrt {a}}} \right ) b{k}^{2}\sqrt {\pi }+\sqrt {2}{{\rm e}^{1/2\,{\frac {2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac {\sqrt {2} \left ( ax+k \right ) }{\sqrt {a}}} \right ) ab\sqrt {\pi }-2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}y{a}^{5/2}-2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}{a}^{3/2}bx-2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}{a}^{3/2}c+2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}bk\sqrt {a} \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (y^2-a^2 x^2+3 a) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a]; pde = D[w[x, y], x] + (y^2 - a^2*x^2 + 3*a)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-y \text {ParabolicCylinderD}\left (-2,i \sqrt {2} \sqrt {a} x\right )+a x \text {ParabolicCylinderD}\left (-2,i \sqrt {2} \sqrt {a} x\right )+i \sqrt {2} \sqrt {a} \text {ParabolicCylinderD}\left (-1,i \sqrt {2} \sqrt {a} x\right )}{y \text {ParabolicCylinderD}\left (1,\sqrt {2} \sqrt {a} x\right )+a x \text {ParabolicCylinderD}\left (1,\sqrt {2} \sqrt {a} x\right )-\sqrt {2} \sqrt {a} \text {ParabolicCylinderD}\left (2,\sqrt {2} \sqrt {a} x\right )}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a'; pde:=diff(w(x,y),x)+(y^2-a^2*x^2+3*a)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {-a{x}^{2}+yx+1}{\sqrt {\pi }\erf \left ( \sqrt {-a}x \right ) \left ( -a \right ) ^{3/2}{x}^{2}+\sqrt {\pi }\erf \left ( \sqrt {-a}x \right ) \sqrt {-a}yx-{{\rm e}^{a{x}^{2}}}ax+\sqrt {\pi }\erf \left ( \sqrt {-a}x \right ) \sqrt {-a}+{{\rm e}^{a{x}^{2}}}y}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (y^2-a^2 x^2+a) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a]; pde = D[w[x, y], x] + (y^2 - a^2*x^2 + a)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {-\sqrt {\pi } y \text {Erfi}\left (\sqrt {a} x\right )+\sqrt {\pi } a x \text {Erfi}\left (\sqrt {a} x\right )-2 \sqrt {a} e^{a x^2}}{2 \sqrt {a} (a x-y)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a'; pde:=diff(w(x,y),x)+(y^2-a^2*x^2+a)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( -ax+y \right ) \sqrt {\pi }}{\sqrt {\pi }\erf \left ( \sqrt {-a}x \right ) ax-\sqrt {\pi }\erf \left ( \sqrt {-a}x \right ) y-2\,\sqrt {-a}{{\rm e}^{a{x}^{2}}}}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (y^2+a x y+a) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a]; pde = D[w[x, y], x] + (y^2 + a*x*y + a)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {2 \pi } \sqrt {a} x y \text {Erfi}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )+\sqrt {2 \pi } \sqrt {a} \text {Erfi}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )-2 y e^{\frac {a x^2}{2}}}{2 \sqrt {2} \sqrt {a} (x y+1)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a'; pde:=diff(w(x,y),x)+(y^2+a*x*y+a)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{yx+1} \left ( -\erf \left ( 1/2\,\sqrt {-2\,a}x \right ) yax+{{\rm e}^{1/2\,a{x}^{2}}}\sqrt {-2\,{\frac {a}{\pi }}}y-a\erf \left ( 1/2\,\sqrt {-2\,a}x \right ) \right ) {\frac {1}{\sqrt {-2\,{\frac {a}{\pi }}}}}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (y^2+a x y-a b x-b^2) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = D[w[x, y], x] + (y^2 + a*x*y - a*b*x - b^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {2 b^2}{a}} \left (2 \sqrt {a} e^{\frac {2 b^2}{a}+\frac {a x^2}{2}+2 b x}+\sqrt {2 \pi } y \text {Erfi}\left (\frac {\sqrt {2} b}{\sqrt {a}}+\frac {\sqrt {a} x}{\sqrt {2}}\right )-\sqrt {2 \pi } b \text {Erfi}\left (\frac {\sqrt {2} b}{\sqrt {a}}+\frac {\sqrt {a} x}{\sqrt {2}}\right )\right )}{2 \sqrt {a} (b-y)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde:=diff(w(x,y),x)+(y^2+a*x*y-a*b*x-b^2)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {\sqrt {2}}{\sqrt {-a} \left ( b-y \right ) } \left ( {{\rm e}^{1/2\,{\frac { \left ( ax+2\,b \right ) ^{2}}{a}}}}\sqrt {2}\sqrt {-a}+\sqrt {\pi }\erf \left ( 1/2\,{\frac { \left ( ax+2\,b \right ) \sqrt {2}}{\sqrt {-a}}} \right ) b-\sqrt {\pi }\erf \left ( 1/2\,{\frac { \left ( ax+2\,b \right ) \sqrt {2}}{\sqrt {-a}}} \right ) y \right ) {{\rm e}^{-2\,{\frac {{b}^{2}}{a}}}}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + k(a x+b y+c)^2 w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c, k]; pde = D[w[x, y], x] + k*(a*x + a*y + c)^2*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-2 i a \sqrt {k} x} \left (i a \sqrt {k} x+i a \sqrt {k} y+i c \sqrt {k}+1\right )}{2 a \sqrt {k} \left (a \sqrt {k} x+a \sqrt {k} y+c \sqrt {k}+i\right )}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c';k:='k'; pde:=diff(w(x,y),x)+k*(a*x+a*y+c)^2*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {xa\sqrt {k}-\arctan \left ( ya\sqrt {k}+xa\sqrt {k}+\sqrt {k}c \right ) }{a\sqrt {k}}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + (a y^2+c x^2+y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, c]; pde = x*D[w[x, y], x] + (a*y^2 + c*x^2 + y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {a} y}{\sqrt {c} x}\right )}{\sqrt {a}}-c x}{c}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';A:='A'; C:='C'; pde:=x*diff(w(x,y),x)+(a*y^2+c*x^2+y)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {ca}} \left ( -x\sqrt {ca}+\arctan \left ( {\frac {ya}{x\sqrt {ca}}} \right ) \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + (a y^2+b x y+c x^2 + y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, c, b]; pde = x*D[w[x, y], x] + (a*y^2 + b*x*y + c*x^2 + y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {b x \sqrt {4 a c-b^2}+2 a y \sqrt {4 a c-b^2}}{4 a c x-b^2 x}\right )-4 a c x+b^2 x}{4 a c-b^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';A:='A'; C:='C';b:='b'; pde:=x*diff(w(x,y),x)+(a*y^2+b*x*y+c*x^2+y)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {4\,ca-{b}^{2}}} \left ( -x\sqrt {4\,ca-{b}^{2}}+2\,\arctan \left ( {\frac {2\,ya+bx}{x\sqrt {4\,ca-{b}^{2}}}} \right ) \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x + c) w_x + \left ( \alpha (a y+b x)^2+\beta ( a y+b x) - b x+\gamma \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, c, b, alpha, gamma, beta]; pde = (a*x + c)*D[w[x, y], x] + (alpha*(a*y + b*x)^2 + beta*(a*y + b*x) - b*x + gamma)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \left (2 \tan ^{-1}\left (\frac {2 a^3 \alpha ^2 y \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}+2 a^2 \alpha ^2 b x \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}+a^2 \alpha \beta \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}}{4 a \alpha \gamma -a \beta ^2+4 \alpha b c}\right )-a \alpha \log (a x+c) \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';A:='A'; C:='C';b:='b';alpha:='alpha';g:='g';beta:='beta'; pde := (a*x + c)*diff(w(x,y),x)+(alpha*(a*y+b*x)^2+beta*(a*y+b*x)-b*x+g)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {{a}^{3} \left ( 4\,ga\alpha -a{\beta }^{2}+4\,\alpha \,bc \right ) }} \left ( -2\,\arctan \left ( {\frac {{a}^{2} \left ( 2\,ya\alpha +2\,\alpha \,bx+\beta \right ) }{\sqrt {4\,{a}^{4}\alpha \,g-{a}^{4}{\beta }^{2}+4\,{a}^{3}\alpha \,bc}}} \right ) {a}^{2}+\ln \left ( ax+c \right ) \sqrt {{a}^{3} \left ( 4\,ga\alpha -a{\beta }^{2}+4\,\alpha \,bc \right ) } \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^2 w_x + b y^2 w_y =0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = a*x^2*D[w[x, y], x] + b*y^2*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {b y-a x}{a x y}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := a*x^2*diff(w(x,y),x)+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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {-ax+by}{axy}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^2+b) w_x - \left ( y^2-2 x y+(1-a)x^2 -b \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = (a*x^2 + b)*D[w[x, y], x] - (y^2 - 2*x*y + (1 - a)*x^2 - b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {y \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )-x \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )-\sqrt {a} \sqrt {b}}{\sqrt {a} \sqrt {b} x-\sqrt {a} \sqrt {b} y}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := (a*x^2+b)*diff(w(x,y),x)-(y^2-2*x*y+(1-a)*x^2-b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {ab} \left ( x-y \right ) } \left ( \arctan \left ( {\frac {ax}{\sqrt {ab}}} \right ) y-\arctan \left ( {\frac {ax}{\sqrt {ab}}} \right ) x-\sqrt {ab} \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x^2+b_1 x + x_1) w_x + (a_2 y^2+b_2 y+c_2) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a1, b1, c1, a2, b2, c2]; pde = (a1*x^2 + b1*x + c1)*D[w[x, y], x] + (a2*y^2 + b2*y + c2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 \left (\frac {4 \text {a2} \text {c2} \sqrt {4 \text {a1} \text {c1}-\text {b1}^2} \tan ^{-1}\left (\frac {2 \text {a1} x+\text {b1}}{\sqrt {4 \text {a1} \text {c1}-\text {b1}^2}}\right )}{\text {b1}^2-4 \text {a1} \text {c1}}-\frac {\text {b2}^2 \sqrt {4 \text {a1} \text {c1}-\text {b1}^2} \tan ^{-1}\left (\frac {2 \text {a1} x+\text {b1}}{\sqrt {4 \text {a1} \text {c1}-\text {b1}^2}}\right )}{\text {b1}^2-4 \text {a1} \text {c1}}+\sqrt {4 \text {a2} \text {c2}-\text {b2}^2} \tan ^{-1}\left (\frac {2 \text {a2} y \sqrt {4 \text {a2} \text {c2}-\text {b2}^2}+\text {b2} \sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}{4 \text {a2} \text {c2}-\text {b2}^2}\right )\right )}{4 \text {a2} \text {c2}-\text {b2}^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a1:='a1';b1:='b1';c1:='c1';a2:='a2';b2:='b2';c2:='c2'; pde := (a1*x^2+b1*x+c1)*diff(w(x,y),x)+ (a2*y^2+b2*y+c2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\frac {1}{\sqrt {4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}}\sqrt {4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}} \left ( \sqrt {4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}\arctan \left ( {\frac {2\,{\it a1}\,x+{\it b1}}{\sqrt {4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}}}} \right ) -\arctan \left ( {\frac {2\,{\it a2}\,y+{\it b2}}{\sqrt {4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}}} \right ) \sqrt {4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (x-a)(x-b) w_x - \left ( y^2+k(y+x-a)(y+x-b)\right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = (x - a)*(x - b)*D[w[x, y], x] - (y^2 + k*(y + x - a)*(y + x - b))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \left (2 \tan ^{-1}\left (\frac {-\frac {2 k^2 x \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{(a-b)^2}-\frac {2 k x \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{(a-b)^2}-\frac {2 k^2 y \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{(a-b)^2}-\frac {4 k y \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{(a-b)^2}-\frac {2 y \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{(a-b)^2}+\frac {2 b k^2 \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{(a-b)^2}+\frac {k^2 \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{a-b}+\frac {2 b k \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{(a-b)^2}+\frac {k \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}{a-b}}{k^2}\right )+\frac {k \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \log (x-a)}{a-b}+\frac {\sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \log (x-a)}{a-b}-\frac {k \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \log (x-b)}{a-b}-\frac {\sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \log (x-b)}{a-b}\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := (x-a)*(x-b)*diff(w(x,y),x)- (y^2+k*(y+x-a)*(y+x-b))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac { \left ( b-x \right ) ^{k} \left ( -kb+kx+ky+y \right ) \left ( a-x \right ) ^{-k}}{-ak+kx+ky+y}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 y^2+b_1 y + c_1) w_x +( a_2 x^2+b_2 x+c_2) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a1, b1, c1, a2, b2, c2]; pde = (a1*y^2 + b1*y + c1)*D[w[x, y], x] + (a2*x^2 + b2*x + c2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{6} \left (2 \text {a1} y^3-2 \text {a2} x^3+3 \text {b1} y^2-3 \text {b2} x^2+6 \text {c1} y-6 \text {c2} x\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a1:='a1';b1:='b1';c1:='c1';a2:='a2';b2:='b2';c2:='c2'; pde := (a1*y^2+b1*y+c1)*diff(w(x,y),x)+ (a2*x^2+b2*x+c2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\it a1}\,{y}^{3}-1/3\,{\it a2}\,{x}^{3}+1/2\,{\it b1}\,{y}^{2}-1/2\,{x}^{2}{\it b2}+{\it c1}\,y-{\it c2}\,x \right ) \]
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Added January 2, 2019.
Problem 2.2.2.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ y(a x+b) w_x +( a y^2-c x) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = y*(a*x + b)*D[w[x, y], x] + (a*y^2 - c*x)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a^2 y^2-2 a c x-b c}{a^2 (a x+b)^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde := y*(a*x+b)*diff(w(x,y),x)+ (a*y^2-c*x)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{a}^{2}{y}^{2}-2\,acx-bc}{{a}^{2} \left ( {a}^{2}{x}^{2}+2\,axb+{b}^{2} \right ) }} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y^2+b x) w_x -(c x^2+b y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = (a*y^2 + b*x)*D[w[x, y], x] - (x*x^2 + b*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{12} \left (4 a y^3+12 b x y+3 x^4\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde := (a*y^2+b*x)*diff(w(x,y),x)- (x*x^2+b*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{y}^{3}-1/4\,{x}^{4}-bxy \right ) \]
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Added January 2, 2019.
Problem 2.2.2.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y^2+b x^2) w_x +2 b x w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a, b]; pde = (a*y^2 + b*x^2)*D[w[x, y], x] + 2*b*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'; pde := (a*y^2+b*x^2)*diff(w(x,y),x)+ 2*b*x*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( {y}^{2}a+b{x}^{2}+2\,ya+2\,a \right ) {{\rm e}^{-y}}}{b}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y^2+b x^2) w_x +2 b x y w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = (a*y^2 + b*x^2)*D[w[x, y], x] + 2*b*x*y*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\log \left (\frac {b x^2-a y^2}{y}\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := (a*y^2+b*x^2)*diff(w(x,y),x)+ 2*b*x*y*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{{y}^{2}a-b{x}^{2}}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y^2+x^2) w_x +(b x^2+c-2 x y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = (a*y^2 + x^2)*D[w[x, y], x] + (b*x^2 + c - 2*x*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{3} \left (a y^3-b x^3-3 c x+3 x^2 y\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := (a*y^2+x^2)*diff(w(x,y),x)+(b*x^2+c-2*x*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{y}^{3}+1/3\,b{x}^{3}-y{x}^{2}+cx \right ) \]
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Added January 2, 2019.
Problem 2.2.2.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (A y^2+B x^2-a^2 B) w_x +(C y^2+2 B x y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, A, B, C0]; pde = (A*y^2 + B*x^2 - a^2*B)*D[w[x, y], x] + (C0*y^2 + 2*B*x*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a^2 (-B)-A y^2+B x^2+\text {C0} x y}{y}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';A:='A';B:='B';C:='C'; pde := (A*y^2+B*x^2-a^2*B)*diff(w(x,y),x)+(C*y^2+2*B*x*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {A{y}^{2}+{a}^{2}B-B{x}^{2}-Cyx}{y}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y^2+b x^2+c y) w_x +2 b x w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a, b, c]; pde = (a*y^2 + b*x^2 + c*y)*D[w[x, y], x] + 2*b*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'; pde := (a*y^2+b*x^2+c*y)*diff(w(x,y),x)+2*b*x*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( {y}^{2}a+b{x}^{2}+2\,ya+cy+2\,a+c \right ) {{\rm e}^{-y}}}{b}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (A x y+B x^2+k x) w_x +(D y^2+E x y+F x^2+k y)w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, A, B, D0, E0, F, k]; pde = (A*x*y + B*x^2 + k*x)*D[w[x, y], x] + (D0*y^2 + E0*x*y + F*x^2 + k*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';D0:='D0';E0:='E0';k:='k';F0:='F0'; pde := (A*x*y+B*x^2+k*x)*diff(w(x,y),x)+(D0*y^2+E0*x*y+F0*x^2+k*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { Exception } \] Timed out
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Added January 2, 2019.
Problem 2.2.2.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (A x y+A k y+B x^2+B k x) w_x +(C y^2+D x y+k(D-B)y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, A, B, D0, E0, C0, k]; pde = (A*x*y + A*k*y + B*x^2 + B*k*x)*D[w[x, y], x] + (C0*y^2 + D0*x*y + k*(D0 - B)*y)*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';D0:='D0';E0:='E0';k:='k';C0:='C0'; pde := (A*x*y+A*k*y+B*x^2+B*k*x)*diff(w(x,y),x)+(C0*y^2+D0*x*y+k*(D0-B)*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
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Added January 2, 2019.
Problem 2.2.2.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (A y^2+B x y+C x^2+k x) w_x +(D y^2+E x y + F x^2+k y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, A, B, D0, E0, C0, k, F0]; pde = (A*y^2 + B*x*y + C0*x^2 + k*x)*D[w[x, y], x] + (D0*y^2 + E0*x*y + F0*x^2 + k*y)*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';D0:='D0';E0:='E0';k:='k';C0:='C0'; pde := (A*y^2+B*x*y+C0*x^2+k*x)*diff(w(x,y),x)+(D0*y^2+E0*x*y+F0*x^2+k*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
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Added January 2, 2019.
Problem 2.2.2.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (A y^2+B x y+C x^2) w_x +(D y^2+E x y + F x^2) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, A, B, D0, E0, C0, F0]; pde = (A*y^2 + B*x*y + C0*x^2)*D[w[x, y], x] + (D0*y^2 + E0*x*y + F0*x^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';D0:='D0';E0:='E0';C0:='C0'; pde := (A*y^2+B*x*y+C0*x^2)*diff(w(x,y),x)+(D0*y^2+E0*x*y+F0*x^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\sum _{{\it \_R}=\RootOf \left ( A{{\it \_Z}}^{3}+ \left ( B-{\it D0} \right ) {{\it \_Z}}^{2}+ \left ( {\it C0}-{\it E0} \right ) {\it \_Z}-{\it F0} \right ) }{\frac {A{{\it \_R}}^{2}+B{\it \_R}+{\it C0}}{3\,A{{\it \_R}}^{2}+2\,B{\it \_R}-2\,{\it D0}\,{\it \_R}+{\it C0}-{\it E0}}\ln \left ( {\frac {-{\it \_R}\,x+y}{x}} \right ) }-\ln \left ( x \right ) \right ) \] solution contains RootOf
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Added January 2, 2019.
Problem 2.2.2.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (A y^2+2 B x y+D x^2+a) w_x -(D y^2+2 D x y-E x^2-b) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, A, B, D0, E0, b, a]; pde = (A*y^2 + 2*B*x*y + D0*x^2 + a)*D[w[x, y], x] - (D0*y^2 + 2*D0*x*y - E0*x^2 - b)*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';D0:='D0';E0:='E0';a:='a';b:='b'; pde := (A*y^2+2*B*x*y+D0*x^2+a)*diff(w(x,y),x)-(D0*y^2+2*D0*x*y-E0*x^2-b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
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Added January 2, 2019.
Problem 2.2.2.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (y^2-2 x y+x^2+a y) w_x +a y w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a]; pde = (y^2 - 2*x*y + x^2 + a*y)*D[w[x, y], x] + a*y*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'; pde := (y^2-2*x*y+x^2+a*y)*diff(w(x,y),x)+a*y*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {\ln \left ( y \right ) x-\ln \left ( y \right ) y+a}{x-y}} \right ) \]
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Added January 2, 2019.
Problem 2.2.2.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke (1965).
Solve for \(w(x,y)\)
\[ (x f_1-f_2) w_x +(y f_1-f_3)w_y = 0 \] Where \(f_n = a_n+b_n x + c_n y\).
Mathematica ✗
ClearAll[w, x, y, a1, a2, a3, b1, b2, b3, c1, c2, c3]; pde = (x*(a1 + b1*x + c1*y) - (a2 + b2*x + c2*y))*D[w[x, y], x] + (y*(a1 + b1*x + c1*y) - (a3 + b3*x + c3*y))*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';a1:='a1';a2:='a2';a3:='a3';b1:='b1';b2:='b2';b3:='b3';c1:='c1';c2:='c2';c3:='c3'; pde := (x *(a1+b1*x+c1*y)-(a2+b2*x+c2*y))*diff(w(x,y),x)+(y*(a1+b1*x+c1*y)-(a3+b3*x+c3*y))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text {Too large to display} \]