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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{x}^{3}-1/2\,{x}^{2}b-cx+y \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt {4\,ca-{b}^{2}}} \left ( x\sqrt {4\,ca-{b}^{2}}-2\,\arctan \left ( {\frac {2\,ya+b}{\sqrt {4\,ca-{b}^{2}}}} \right ) \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{{\rm e}^{-ax}} \left ( y{a}^{3}+x \left ( xb+c \right ) {a}^{2}+ \left ( 2\,xb+c \right ) a+2\,b \right ) }{{a}^{3}}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {{{\rm e}^{-x \left ( ax+2\,k \right ) }}}{{a}^{5/2}} \left ( \sqrt {\pi }\erf \left ( 1/2\,{\frac {\sqrt {2} \left ( ax+k \right ) }{\sqrt {a}}} \right ) \sqrt {2} \left ( \left ( {k}^{2}+a \right ) b+{a}^{2}s-cak \right ) {{\rm e}^{1/2\,{\frac {2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}+2\, \left ( \left ( -xb-c \right ) {a}^{3/2}+kb\sqrt {a}-y{a}^{5/2} \right ) {{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }} \right ) } \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {D_{-2}\left (i \sqrt {2} \sqrt {a} x\right ) (a x-y)+i \sqrt {2} \sqrt {a} D_{-1}\left (i \sqrt {2} \sqrt {a} x\right )}{D_1\left (\sqrt {2} \sqrt {a} x\right ) (a x+y)-\sqrt {2} \sqrt {a} D_2\left (\sqrt {2} \sqrt {a} x\right )}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-a{x}^{2}+xy+1}{ \left ( ax-y \right ) {{\rm e}^{a{x}^{2}}}-\sqrt {\pi }\erf \left ( \sqrt {-a}x \right ) \left ( {x}^{2} \left ( -a \right ) ^{3/2}+xy\sqrt {-a}+\sqrt {-a} \right ) }} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {a} x\right ) (y-a x)+2 \sqrt {a} e^{a x^2}}{2 \sqrt {a} (a x-y)}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac { \left ( ax-y \right ) \sqrt {\pi }}{-2\,\sqrt {-a}{{\rm e}^{a{x}^{2}}}+\sqrt {\pi }\erf \left ( \sqrt {-a}x \right ) \left ( ax-y \right ) }} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )-\frac {y e^{\frac {a x^2}{2}}}{\sqrt {2} \sqrt {a} (x y+1)}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{xy+1} \left ( -{{\rm e}^{1/2\,a{x}^{2}}}\sqrt {-2\,{\frac {a}{\pi }}}y+a\erf \left ( 1/2\,\sqrt {-2\,a}x \right ) \left ( xy+1 \right ) \right ) {\frac {1}{\sqrt {-2\,{\frac {a}{\pi }}}}}} \right ) \]
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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["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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {2 b^2}{a}} \left (\sqrt {2 \pi } (y-b) \text {Erfi}\left (\frac {a x+2 b}{\sqrt {2} \sqrt {a}}\right )+2 \sqrt {a} e^{\frac {(a x+2 b)^2}{2 a}}\right )}{2 \sqrt {a} (b-y)}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\sqrt {2}}{\sqrt {-a} \left ( 2\,b-2\,y \right ) } \left ( \sqrt {\pi } \left ( b-y \right ) \erf \left ( 1/2\,{\frac { \left ( ax+2\,b \right ) \sqrt {2}}{\sqrt {-a}}} \right ) +{{\rm e}^{1/2\,{\frac { \left ( ax+2\,b \right ) ^{2}}{a}}}}\sqrt {2}\sqrt {-a} \right ) {{\rm e}^{-2\,{\frac {{b}^{2}}{a}}}}} \right ) \]
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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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {xa\sqrt {k}-\arctan \left ( \sqrt {k} \left ( \left ( x+y \right ) a+c \right ) \right ) }{a\sqrt {k}}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {ca}} \left ( -x\sqrt {ca}+\arctan \left ( {\frac {ya}{x\sqrt {ca}}} \right ) \right ) } \right ) \]
____________________________________________________________________________________
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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt {4\,ca-{b}^{2}}} \left ( x\sqrt {4\,ca-{b}^{2}}-2\,\arctan \left ( {\frac {2\,ya+xb}{x\sqrt {4\,ca-{b}^{2}}}} \right ) \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt { \left ( 4\,g\alpha -{\beta }^{2} \right ) {a}^{4}+4\,{a}^{3}\alpha \,bc}} \left ( -2\,\arctan \left ( {\frac {{a}^{2} \left ( 2\,ya\alpha +2\,\alpha \,bx+\beta \right ) }{\sqrt {4\,{a}^{4}\alpha \,g-{a}^{4}{\beta }^{2}+4\,{a}^{3}\alpha \,bc}}} \right ) {a}^{2}+\ln \left ( ax+c \right ) \sqrt { \left ( 4\,g\alpha -{\beta }^{2} \right ) {a}^{4}+4\,{a}^{3}\alpha \,bc} \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ax-by}{axy}} \right ) \]
____________________________________________________________________________________
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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt {ab} \left ( x-y \right ) } \left ( \left ( x-y \right ) \arctan \left ( {\frac {ax}{\sqrt {ab}}} \right ) +\sqrt {ab} \right ) } \right ) \]
____________________________________________________________________________________
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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\frac {1}{\sqrt {4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}}\sqrt {4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}} \left ( \arctan \left ( {\frac {2\,{\it a1}\,x+{\it b1}}{\sqrt {4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}}}} \right ) \sqrt {4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}-\arctan \left ( {\frac {2\,{\it a2}\,y+{\it b2}}{\sqrt {4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}}} \right ) \sqrt {4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}} \right ) } \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac { \left ( \left ( a-x-y \right ) k-y \right ) \left ( -x+a \right ) ^{k} \left ( b-x \right ) ^{-k}}{k \left ( b-x-y \right ) -y}} \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\it a1}\,{y}^{3}-1/3\,{\it a2}\,{x}^{3}+1/2\,{\it b1}\,{y}^{2}-1/2\,{x}^{2}{\it b2}+{\it c1}\,y-{\it c2}\,x \right ) \]
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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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{y}^{2}{a}^{2}-2\,acx-bc}{{a}^{2} \left ( ax+b \right ) ^{2}}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{y}^{3}-1/4\,{x}^{4}-bxy \right ) \]
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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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{{\rm e}^{-y}} \left ( a \left ( {y}^{2}+2\,y+2 \right ) +{x}^{2}b \right ) }{b}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{{y}^{2}a-{x}^{2}b}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{y}^{3}+1/3\,{x}^{3}b-{x}^{2}y+cx \right ) \]
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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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-A{y}^{2}-{a}^{2}B+B{x}^{2}+Cyx}{y}} \right ) \]
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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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( {y}^{2}a+ \left ( c+2\,a \right ) y+{x}^{2}b+2\,a+c \right ) {{\rm e}^{-y}}}{b}} \right ) \]
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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["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]];
$Aborted
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\sum _{{\it \_R}=\RootOf \left ( A{{\it \_Z}}^{3}+ \left ( B-{\it D0} \right ) {{\it \_Z}}^{2}+ \left ( {\it C0}-{\it E0} \right ) {\it \_Z}-{\it F0} \right ) }{\frac {A{{\it \_R}}^{2}+B{\it \_R}+{\it C0}}{3\,A{{\it \_R}}^{2}+2\, \left ( B-{\it D0} \right ) {\it \_R}+{\it C0}-{\it E0}}\ln \left ( {\frac {-{\it \_R}\,x+y}{x}} \right ) }-\ln \left ( x \right ) \right ) \] 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["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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( x-y \right ) \ln \left ( y \right ) +a}{x-y}} \right ) \]
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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["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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