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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a x^3}{3}-\frac {b x^2}{2}-c x+y\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {1}{3} a \,x^{3}-\frac {1}{2} b \,x^{2}-c x +y \right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \tan ^{-1}\left (\frac {2 a y+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-x\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\sqrt {4 c a -b^{2}}\, x -2 \arctan \left (\frac {2 y a +b}{\sqrt {4 c a -b^{2}}}\right )}{\sqrt {4 c a -b^{2}}}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-a x} \left (b \left (a^2 x^2+2 a x+2\right )+a \left (a^2 y+a c x+c\right )\right )}{a^3}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (a^{3} y +\left (b x +c \right ) a^{2} x +\left (2 b x +c \right ) a +2 b \right ) {\mathrm e}^{-a x}}{a^{3}}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {1}{2} x (a x+2 k)} \left (2 \sqrt {a} \left (a^2 y+a (b x+c)-b k\right )-\sqrt {2 \pi } e^{\frac {(a x+k)^2}{2 a}} \text {erf}\left (\frac {a x+k}{\sqrt {2} \sqrt {a}}\right ) \left (a^2 s+a (b-c k)+b k^2\right )\right )}{2 a^{5/2}}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (\sqrt {\pi }\, \sqrt {2}\, \left (a^{2} s -a c k +\left (k^{2}+a \right ) b \right ) \erf \left (\frac {\sqrt {2}\, \left (a x +k \right )}{2 \sqrt {a}}\right ) {\mathrm e}^{\frac {2 a^{2} x^{2}+4 k x a +k^{2}}{2 a}}+2 \left (-a^{\frac {5}{2}} y +\sqrt {a}\, b k +\left (-b x -c \right ) a^{\frac {3}{2}}\right ) {\mathrm e}^{\frac {\left (a x +2 k \right ) x}{2}}\right ) {\mathrm e}^{-\left (a x +2 k \right ) x}}{2 a^{\frac {5}{2}}}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {(a x-y) \operatorname {ParabolicCylinderD}\left (-2,i \sqrt {2} \sqrt {a} x\right )+i \sqrt {2} \sqrt {a} \operatorname {ParabolicCylinderD}\left (-1,i \sqrt {2} \sqrt {a} x\right )}{(a x+y) \operatorname {ParabolicCylinderD}\left (1,\sqrt {2} \sqrt {a} x\right )-\sqrt {2} \sqrt {a} \operatorname {ParabolicCylinderD}\left (2,\sqrt {2} \sqrt {a} x\right )}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-a \,x^{2}+x y +1}{-\sqrt {\pi }\, \left (\left (-a \right )^{\frac {3}{2}} x^{2}+\sqrt {-a}\, x y +\sqrt {-a}\right ) \erf \left (\sqrt {-a}\, x \right )+\left (a x -y \right ) {\mathrm e}^{a \,x^{2}}}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\pi } (y-a x) \operatorname {Erfi}\left (\sqrt {a} x\right )+2 \sqrt {a} e^{a x^2}}{2 \sqrt {a} (a x-y)}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (a x -y \right ) \sqrt {\pi }}{\sqrt {\pi }\, \left (a x -y \right ) \erf \left (\sqrt {-a}\, x \right )-2 \sqrt {-a}\, {\mathrm e}^{a \,x^{2}}}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \sqrt {\pi } \operatorname {Erfi}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )-\frac {y e^{\frac {a x^2}{2}}}{\sqrt {2} \sqrt {a} (x y+1)}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (x y +1\right ) a \erf \left (\frac {\sqrt {-2 a}\, x}{2}\right )-\sqrt {-\frac {2 a}{\pi }}\, y \,{\mathrm e}^{\frac {a \,x^{2}}{2}}}{\sqrt {-\frac {2 a}{\pi }}\, \left (x y +1\right )}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {2 b^2}{a}} \left (\sqrt {2 \pi } (y-b) \operatorname {Erfi}\left (\frac {a x+2 b}{\sqrt {2} \sqrt {a}}\right )+2 \sqrt {a} e^{\frac {(a x+2 b)^2}{2 a}}\right )}{2 \sqrt {a} (b-y)}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (\sqrt {\pi }\, \left (b -y \right ) \erf \left (\frac {\left (a x +2 b \right ) \sqrt {2}}{2 \sqrt {-a}}\right )+\sqrt {2}\, \sqrt {-a}\, {\mathrm e}^{\frac {\left (a x +2 b \right )^{2}}{2 a}}\right ) \sqrt {2}\, {\mathrm e}^{-\frac {2 b^{2}}{a}}}{\sqrt {-a}\, \left (2 b -2 y \right )}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-2 i a \sqrt {k} x} \left (i a \sqrt {k} (x+y)+i c \sqrt {k}+1\right )}{2 a \sqrt {k} \left (a \sqrt {k} (x+y)+c \sqrt {k}+i\right )}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a \sqrt {k}\, x -\arctan \left (\left (\left (x +y \right ) a +c \right ) \sqrt {k}\right )}{a \sqrt {k}}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\tan ^{-1}\left (\frac {\sqrt {a} y}{\sqrt {c} x}\right )}{\sqrt {a} \sqrt {c}}-x\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\sqrt {a c}\, x +\arctan \left (\frac {a y}{\sqrt {a c}\, x}\right )}{\sqrt {a c}}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \tan ^{-1}\left (\frac {2 a y+b x}{x \sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-x\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\sqrt {4 a c -b^{2}}\, x -2 \arctan \left (\frac {2 y a +b x}{\sqrt {4 a c -b^{2}}\, x}\right )}{\sqrt {4 a c -b^{2}}}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \left (2 \tan ^{-1}\left (\frac {2 \alpha (a y+b x)+\beta }{a \alpha \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}}\right )-a \alpha \log (a x+c) \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}\right )\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-2 a^{2} \arctan \left (\frac {\left (2 a \alpha y +2 \alpha b x +\beta \right ) a^{2}}{\sqrt {4 a^{4} \alpha g -a^{4} \beta ^{2}+4 a^{3} \alpha b c}}\right )+\sqrt {4 a^{3} \alpha b c +\left (4 g \alpha -\beta ^{2}\right ) a^{4}}\, \ln \left (a x +c \right )}{\sqrt {4 a^{3} \alpha b c +\left (4 g \alpha -\beta ^{2}\right ) a^{4}}}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b}{a x}-\frac {1}{y}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a x -b y}{a x y}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\frac {(y-x) \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}-1}{x-y}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (x -y \right ) \arctan \left (\frac {a x}{\sqrt {a b}}\right )+\sqrt {a b}}{\sqrt {a b}\, \left (x -y \right )}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \tan ^{-1}\left (\frac {2 \text {a2} y+\text {b2}}{\sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}\right )}{\sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}-\frac {2 \tan ^{-1}\left (\frac {2 \text {a1} x+\text {b1}}{\sqrt {4 \text {a1} \text {c1}-\text {b1}^2}}\right )}{\sqrt {4 \text {a1} \text {c1}-\text {b1}^2}}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {2 \left (\sqrt {4 \mathit {c2} \mathit {a2} -\mathit {b2}^{2}}\, \arctan \left (\frac {2 \mathit {a1} x +\mathit {b1}}{\sqrt {4 \mathit {c1} \mathit {a1} -\mathit {b1}^{2}}}\right )-\sqrt {4 \mathit {c1} \mathit {a1} -\mathit {b1}^{2}}\, \arctan \left (\frac {2 \mathit {a2} y +\mathit {b2}}{\sqrt {4 \mathit {c2} \mathit {a2} -\mathit {b2}^{2}}}\right )\right )}{\sqrt {4 \mathit {c1} \mathit {a1} -\mathit {b1}^{2}}\, \sqrt {4 \mathit {c2} \mathit {a2} -\mathit {b2}^{2}}}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {(k+1) \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} (\log (x-a)-\log (x-b))}{2 (a-b)}-\tan ^{-1}\left (\frac {a k+b k-2 (k (x+y)+y)}{(k+1) \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}\right )\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (\left (b -x -y \right ) k -y \right ) \left (a -x \right )^{-k} \left (b -x \right )^{k}}{\left (a -x -y \right ) k -y}\right )\]
____________________________________________________________________________________
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["Global`*"]; 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 ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {1}{3} \mathit {a1} \,y^{3}-\frac {1}{3} \mathit {a2} \,x^{3}+\frac {1}{2} \mathit {b1} \,y^{2}-\frac {1}{2} \mathit {b2} \,x^{2}+\mathit {c1} y -\mathit {c2} x \right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {a \left (a y^2-2 c x\right )-b c}{a^2 (a x+b)^2}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a^{2} y^{2}-2 a c x -b c}{\left (a x +b \right )^{2} a^{2}}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {a y^3}{3}+b x y+\frac {x^4}{4}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {1}{3} a \,y^{3}-\frac {1}{4} x^{4}-b x y \right )\]
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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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (b \,x^{2}+\left (y^{2}+2 y +2\right ) a \right ) {\mathrm e}^{-y}}{b}\right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\log \left (\frac {b x^2}{y}-a y\right )\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {y}{y^{2} a -b \,x^{2}}\right )\]
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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["Global`*"]; 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 ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {1}{3} a \,y^{3}+\frac {1}{3} b \,x^{3}-x^{2} y +c x \right )\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {a^2 (-B)+y (\text {C0} x-A y)+B x^2}{y}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-A \,y^{2}-a^{2} B +B \,x^{2}+C x y}{y}\right )\]
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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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (a \,y^{2}+b \,x^{2}+2 a +c +\left (2 a +c \right ) y \right ) {\mathrm e}^{-y}}{b}\right )\]
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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["Global`*"]; 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]];
$Aborted
Maple ✗
restart; 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'));
time expired
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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["Global`*"]; 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]];
Failed
Maple ✗
restart; 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'));
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["Global`*"]; 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]];
Failed
Maple ✗
restart; 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'));
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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\ln \left (x \right )-\frac {\left (A \RootOf \left (A \,\textit {\_Z}^{3}+\left (B -\mathit {D0} \right ) \textit {\_Z}^{2}-\mathit {F0} +\left (\mathit {C0} -\mathit {E0} \right ) \textit {\_Z} \right )^{2}+B \RootOf \left (A \,\textit {\_Z}^{3}+\left (B -\mathit {D0} \right ) \textit {\_Z}^{2}-\mathit {F0} +\left (\mathit {C0} -\mathit {E0} \right ) \textit {\_Z} \right )+\mathit {C0} \right ) \ln \left (\frac {-\RootOf \left (A \,\textit {\_Z}^{3}+\left (B -\mathit {D0} \right ) \textit {\_Z}^{2}-\mathit {F0} +\left (\mathit {C0} -\mathit {E0} \right ) \textit {\_Z} \right ) x +y}{x}\right )}{3 A \RootOf \left (A \,\textit {\_Z}^{3}+\left (B -\mathit {D0} \right ) \textit {\_Z}^{2}-\mathit {F0} +\left (\mathit {C0} -\mathit {E0} \right ) \textit {\_Z} \right )^{2}+\mathit {C0} -\mathit {E0} +2 \left (B -\mathit {D0} \right ) \RootOf \left (A \,\textit {\_Z}^{3}+\left (B -\mathit {D0} \right ) \textit {\_Z}^{2}-\mathit {F0} +\left (\mathit {C0} -\mathit {E0} \right ) \textit {\_Z} \right )}\right )\] 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["Global`*"]; 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]];
Failed
Maple ✗
restart; 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'));
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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a +\left (x -y \right ) \ln \left (y \right )}{x -y}\right )\]
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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["Global`*"]; 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]];
Failed
Maple ✓
restart; 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'));
\[\text {Expression too large to display}\]
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