Added Feb. 9, 2019.
Problem Chapter 3.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a \sqrt {x^2+y^2} \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*Sqrt[x^2 + y^2]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to a \sqrt {x^2+y^2}+c_1\left (\frac {y}{x}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x) + y*diff(w(x,y),y) =a*sqrt(x^2+y^2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =a\sqrt {{x}^{2}+{y}^{2}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
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Added Feb. 9, 2019.
Problem Chapter 3.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = c x y^2+d x^2 y+k \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x*y^2 + d*x^2*y + k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {a \left (2 a^2+5 a b+2 b^2\right ) c_1\left (y x^{-\frac {b}{a}}\right )+k \left (2 a^2+5 a b+2 b^2\right ) \log (x)+a x y (2 a c y+a d x+b c y+2 b d x)}{a (2 a+b) (a+2 b)}\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x) + b*y*diff(w(x,y),y) =c*x*y^2+d*x^2*y+k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {d{x}^{2}y}{2\,a+b}}+{\frac {cx{y}^{2}}{a+2\,b}}+{\frac {k\ln \left ( x \right ) }{a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]
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Added Feb. 9, 2019.
Problem Chapter 3.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + b x w_y = c x y^2+d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == c*x*y^2 + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{w(x,y)\to -\frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )+c y^2 \sqrt {a y^2}}{3 \sqrt {a} b}+c_1\left (\frac {a y^2-b x^2}{2 a}\right )\right \}\\& \left \{w(x,y)\to \frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )+c y^2 \sqrt {a y^2}}{3 \sqrt {a} b}+c_1\left (\frac {a y^2-b x^2}{2 a}\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := a*y*diff(w(x,y),x) + b*x*diff(w(x,y),y) =c*x*y^2+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{{a}^{2}b} \left ( d\ln \left ( {abx{\frac {1}{\sqrt {ab}}}}+\sqrt {{a}^{2}{y}^{2}} \right ) {a}^{2}b+ \left ( {\it \_F1} \left ( {\frac {{y}^{2}a-{x}^{2}b}{a}} \right ) {a}^{2}b+c \left ( \left ( -{\frac {2\,{y}^{2}a}{3}}+{x}^{2}b \right ) \sqrt {{a}^{2}{y}^{2}}+ya \left ( {y}^{2}a-{x}^{2}b \right ) \right ) \right ) \sqrt {ab} \right ) {\frac {1}{\sqrt {ab}}}}\]
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Added Feb. 9, 2019.
Problem Chapter 3.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x +(c y +d) w_y = k x^3+n y^3 \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == k*x^3 + n*y^3; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )-\frac {\log (a x+b) \left (a^3 d^3 n+b^3 c^3 k\right )}{a^4 c^3}+\frac {b^2 k x}{a^3}-\frac {b k x^2}{2 a^2}+\frac {k x^3}{3 a}+\frac {n \left (-3 c^2 d y^2+2 c^3 y^3+6 c d^2 y+11 d^3\right )}{6 c^4}\right \}\right \}\]
Maple ✓
restart; pde := (a*x+b)*diff(w(x,y),x) + (c*y+d)*diff(w(x,y),y) =k*x^3+n*y^3; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{6\,{a}^{4}{c}^{4}} \left ( 6\,{\it \_F1} \left ( {\frac {yc+d}{c} \left ( ax+b \right ) ^{-{\frac {c}{a}}}} \right ) {a}^{4}{c}^{4}+ \left ( -6\,{d}^{3}n{a}^{3}c-6\,{b}^{3}k{c}^{4} \right ) \ln \left ( ax+b \right ) +2\, \left ( \left ( yc+d \right ) n \left ( {c}^{2}{y}^{2}-5/2\,cdy+11/2\,{d}^{2} \right ) {a}^{3}+{a}^{2}{c}^{4}k{x}^{3}-3/2\,ab{c}^{4}k{x}^{2}+3\,{b}^{2}{c}^{4}kx \right ) a \right ) }\]
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Added Feb. 9, 2019.
Problem Chapter 3.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x^2 w_x +x y w_y = y^2 (a x + b y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x^2*D[w[x, y], x] + x*y*D[w[x, y], y] == y^2*(a*x + b*y); sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {y^2 (a x+b y)}{2 x}+c_1\left (\frac {y}{x}\right )\right \}\right \}\]
Maple ✓
restart; pde := x^2*diff(w(x,y),x) + x*y*diff(w(x,y),y) =y^2*(a*x + b*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {{y}^{2}a}{2}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) +{\frac {b{y}^{3}}{2\,x}}\]
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Added Feb. 9, 2019.
Problem Chapter 3.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^3 w_x +b y^3 w_y = c x + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^3*D[w[x, y], x] + b*y^3*D[w[x, y], y] == c*x + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {2 c x+d}{2 a x^2}+c_1\left (\frac {1}{2} \left (\frac {b}{a x^2}-\frac {1}{y^2}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^3*diff(w(x,y),x) + b*y^3*diff(w(x,y),y) =c*x+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =-{\frac {c}{ax}}-{\frac {d}{2\,a{x}^{2}}}+{\it \_F1} \left ( {\frac {a{x}^{2}-b{y}^{2}}{{y}^{2}a{x}^{2}}} \right ) \]
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