Added January 2, 2019.
Problem 2.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +(a \sqrt {x} y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Sqrt[x]*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y e^{-\frac {2}{3} a x^{3/2}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*sqrt(x)*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-2/3\,{x}^{3/2}a}} \right ) \]
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Added January 2, 2019.
Problem 2.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a \sqrt {x} y+ b \sqrt {y}) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Sqrt[x]*y + b*Sqrt[y])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {b \text {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )}{\sqrt [3]{3} a^{2/3}}-\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\\& \left \{w(x,y)\to c_1\left (\frac {b \text {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )}{\sqrt [3]{3} a^{2/3}}+\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*sqrt(x)*y+b*sqrt(y))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/10\,{\frac { \left ( 3\,\sqrt [3]{3} \WhittakerM \left ( 1/3,5/6,1/3\,{x}^{3/2}a \right ) {{\rm e}^{1/6\,{x}^{3/2}a}}bx+5\,bx\sqrt [3]{{x}^{3/2}a}-10\,\sqrt [3]{{x}^{3/2}a}\sqrt {y} \right ) {{\rm e}^{-1/3\,{x}^{3/2}a}}}{\sqrt [3]{{x}^{3/2}a}}} \right ) \]
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Added January 2, 2019.
Problem 2.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +(a \sqrt {x} y+ b x \sqrt {y}) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Sqrt[x]*y + b*x*Sqrt[y])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt [3]{3} b \text {Gamma}\left (\frac {4}{3},\frac {1}{3} a x^{3/2}\right )}{a^{4/3}}+\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*sqrt(x)*y+b*x*sqrt(y))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,{\frac {{{\rm e}^{-1/3\,{x}^{3/2}a}} \left ( 3\,\sqrt [6]{3}\sqrt {x} \WhittakerM \left ( 1/6,2/3,1/3\,{x}^{3/2}a \right ) {{\rm e}^{1/6\,{x}^{3/2}a}}b-4\,\sqrt {y}a\sqrt [6]{{x}^{3/2}a} \right ) }{\sqrt [6]{{x}^{3/2}a}a}} \right ) \]
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Added January 2, 2019.
Problem 2.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +A \sqrt {a x + b y+ c} w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + A*Sqrt[a*x + b*y + c]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{w(x,y)\to c_1\left (x-\frac {a \log \left (\frac {e^{\frac {2 \sqrt {A^2 b^2 (a x+b y+c)}}{a}}}{\left (\sqrt {A^2 b^2 (a x+b y+c)}+a\right )^2}\right )}{A^2 b^2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (x-\frac {a \log \left (\frac {e^{-\frac {2 \sqrt {A^2 b^2 (a x+b y+c)}}{a}}}{\left (a-\sqrt {A^2 b^2 (a x+b y+c)}\right )^2}\right )}{A^2 b^2}\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := diff(w(x,y),x)+ A*sqrt(a*x+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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {x{b}^{2}{A}^{2}-2\,A\sqrt {ax+by+c}b-a\ln \left ( A\sqrt {ax+by+c}b-a \right ) +a\ln \left ( A\sqrt {ax+by+c}b+a \right ) +a\ln \left ( {b}^{2} \left ( ax+by+c \right ) {A}^{2}-{a}^{2} \right ) }{{b}^{2}{A}^{2}}} \right ) \]
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Added January 2, 2019.
Problem 2.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a y + b \sqrt {y^2+c x^2} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*y + b*Sqrt[y^2 + c*x^2])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := x*diff(w(x,y),x)+ ( a*y + b *sqrt(y^2+c*x^2))*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 2.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \left (a x + b \sqrt {y} \right ) w_x - \left ( c \sqrt {x} + a y \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a*x + b*Sqrt[y])*D[w[x, y], x] - (c*Sqrt[x] + a*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {3 a^3 x^3}{8 b^2}+\frac {2}{3} c x^{3/2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (a x y-\frac {2}{3} b y^{3/2}+\frac {2}{3} c x^{3/2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (a x y+\frac {2}{3} b y^{3/2}+\frac {2}{3} c x^{3/2}\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := (a*x+b*sqrt(y))* diff(w(x,y),x)- (c*sqrt(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \RootOf \left ( 8\,{y}^{5/2}ab{c}^{2}+3\,{a}^{4}{y}^{4}-2\,\sqrt [3]{-4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c}{a}^{3}{y}^{3}+3\, \left ( -4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{2/3}{a}^{2}{y}^{2}-4\,x{c}^{2} \left ( -4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{2/3}+12\,a{c}^{2}y{\it \_Z}+ \left ( -4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{4/3}-4\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}acy \right ) \right ) \]
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Added January 2, 2019.
Problem 2.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ \sqrt {f(x)} w_x + \sqrt {f(y)} w_y = 0 \] Where \(f(t) = \sum _{n=0}^{4} a_n t^n \)
Mathematica ✗
ClearAll["Global`*"]; f[t_] := Sum[a[n]*t^n, {n, 1, 4}]; pde = Sqrt[f[x]]*D[w[x, y], x] + Sqrt[f[y]]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; f:=t->sum(a[n]*t^n,n=1..4); pde := sqrt(f(x))* diff(w(x,y),x)+ sqrt(f(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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