6.2.4 2.4
6.2.4.1 [470] problem number 1
problem number 470
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'));
\[w \left (x , y\right ) = f_{1} \left (y \,{\mathrm e}^{-\frac {2 x^{{3}/{2}} a}{3}}\right )\]
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6.2.4.2 [471] problem number 2
problem number 471
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 \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 \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'));
\[w \left (x , y\right ) = f_{1} \left (-\frac {\frac {3 \,{\mathrm e}^{-\frac {x^{{3}/{2}} a}{6}} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {x^{{3}/{2}} a}{3}\right ) 3^{{1}/{3}} b x}{5}+{\mathrm e}^{-\frac {x^{{3}/{2}} a}{3}} \left (x^{{3}/{2}} a \right )^{{1}/{3}} \left (b x -2 \sqrt {y}\right )}{2 \left (x^{{3}/{2}} a \right )^{{1}/{3}}}\right )\]
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6.2.4.3 [472] problem number 3
problem number 472
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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt [3]{3} b \Gamma \left (\frac {4}{3},\frac {1}{3} a x^{3/2}\right )}{a^{4/3}}+\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = f_{1} \left (\frac {-3 \,{\mathrm e}^{-\frac {x^{{3}/{2}} a}{6}} b 3^{{1}/{6}} \sqrt {x}\, \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{{3}/{2}} a}{3}\right )+4 \,{\mathrm e}^{-\frac {x^{{3}/{2}} a}{3}} \left (x^{{3}/{2}} a \right )^{{1}/{6}} \sqrt {y}\, a}{4 \left (x^{{3}/{2}} a \right )^{{1}/{6}} a}\right )\]
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6.2.4.4 [473] problem number 4
problem number 473
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 (\frac {2 a \log \left (e^{-\frac {\sqrt {A^2 b^2 (a x+b y+c)}}{a}} \left (\sqrt {A^2 b^2 (a x+b y+c)}+a\right )\right )}{A^2 b^2}+\frac {c}{a}+x\right )\right \}\\& \left \{w(x,y)\to c_1\left (\frac {2 a \log \left (e^{\frac {\sqrt {A^2 b^2 (a x+b y+c)}}{a}} \left (a-\sqrt {A^2 b^2 (a x+b y+c)}\right )\right )}{A^2 b^2}+\frac {c}{a}+x\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'));
\[w \left (x , y\right ) = f_{1} \left (\frac {x \,A^{2} b^{2}-2 A \sqrt {a x +b y +c}\, b +a \ln \left (A \sqrt {a x +b y +c}\, b +a \right )-a \ln \left (A \sqrt {a x +b y +c}\, b -a \right )+a \ln \left (b^{2} \left (a x +b y +c \right ) A^{2}-a^{2}\right )}{A^{2} b^{2}}\right )\]
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6.2.4.5 [474] problem number 5
problem number 474
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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6.2.4.6 [475] problem number 6
problem number 475
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'));
\[w \left (x , y\right ) = f_{1} \left (\operatorname {RootOf}\left (3 a^{4} y^{4}-3 \left (-4 y^{{3}/{2}} b \,c^{2}-a^{3} y^{3}-6 \textit {\_Z} \,c^{2}+2 \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{{1}/{3}} a^{3} y^{3}+8 a \,y^{{5}/{2}} b \,c^{2}+3 \left (-4 y^{{3}/{2}} b \,c^{2}-a^{3} y^{3}-6 \textit {\_Z} \,c^{2}+2 \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{{2}/{3}} a^{2} y^{2}-4 b \,c^{2} y^{{3}/{2}} \left (-4 y^{{3}/{2}} b \,c^{2}-a^{3} y^{3}-6 \textit {\_Z} \,c^{2}+2 \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{{1}/{3}}+12 a \,c^{2} y \textit {\_Z} -4 x \,c^{2} \left (-4 y^{{3}/{2}} b \,c^{2}-a^{3} y^{3}-6 \textit {\_Z} \,c^{2}+2 \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{{2}/{3}}-6 \textit {\_Z} \,c^{2} \left (-4 y^{{3}/{2}} b \,c^{2}-a^{3} y^{3}-6 \textit {\_Z} \,c^{2}+2 \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{{1}/{3}}-4 \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, a c y +2 c \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, \left (-4 y^{{3}/{2}} b \,c^{2}-a^{3} y^{3}-6 \textit {\_Z} \,c^{2}+2 \sqrt {4 y^{3} b^{2} c^{2}+2 y^{{9}/{2}} a^{3} b +12 y^{{3}/{2}} \textit {\_Z} b \,c^{2}+3 \textit {\_Z} \,a^{3} y^{3}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{{1}/{3}}\right )\right )\]
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6.2.4.7 [476] problem number 7
problem number 476
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]];
\[\left \{\left \{w(x,y)\to -\frac {2 c_1 y \sqrt {x (x (x (a(4) x+a(3))+a(2))+a(1))} \sqrt {\frac {y \text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,1\right ]-1}{y \left (\text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,1\right ]-\text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,3\right ]\right )}} \sqrt {\frac {y \text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,2\right ]-1}{y \left (\text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,2\right ]-\text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,3\right ]\right )}} \left (y \text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,3\right ]-1\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [a(1) \text {$\#$1}^3+a(2) \text {$\#$1}^2+a(3) \text {$\#$1}+a(4)\& ,3\right ]-\frac {1}{y}}{\text {Root}\left [a(1) \text {$\#$1}^3+a(2) \text {$\#$1}^2+a(3) \text {$\#$1}+a(4)\& ,3\right ]-\text {Root}\left [a(1) \text {$\#$1}^3+a(2) \text {$\#$1}^2+a(3) \text {$\#$1}+a(4)\& ,2\right ]}}\right ),\frac {\text {Root}\left [a(1) \text {$\#$1}^3+a(2) \text {$\#$1}^2+a(3) \text {$\#$1}+a(4)\& ,2\right ]-\text {Root}\left [a(1) \text {$\#$1}^3+a(2) \text {$\#$1}^2+a(3) \text {$\#$1}+a(4)\& ,3\right ]}{\text {Root}\left [a(1) \text {$\#$1}^3+a(2) \text {$\#$1}^2+a(3) \text {$\#$1}+a(4)\& ,1\right ]-\text {Root}\left [a(1) \text {$\#$1}^3+a(2) \text {$\#$1}^2+a(3) \text {$\#$1}+a(4)\& ,3\right ]}\right ) \exp \left (\int _1^x-\frac {a(1)+K[1] (2 a(2)+K[1] (3 a(3)+4 a(4) K[1]))}{2 K[1] (a(1)+K[1] (a(2)+K[1] (a(3)+a(4) K[1])))}dK[1]\right )}{\sqrt {y (y (y (a(4) y+a(3))+a(2))+a(1))} \sqrt {\frac {1-y \text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,3\right ]}{y \text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,2\right ]-y \text {Root}\left [\text {$\#$1}^3 a(1)+\text {$\#$1}^2 a(2)+\text {$\#$1} a(3)+a(4)\& ,3\right ]}}}+\int _1^x\exp \left (\int _1^{K[6155]}-\frac {a(1)+K[1] (2 a(2)+K[1] (3 a(3)+4 a(4) K[1]))}{2 K[1] (a(1)+K[1] (a(2)+K[1] (a(3)+a(4) K[1])))}dK[1]\right ) c_1dK[6155]+c_2\right \}\right \}\]
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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