6.4.3 2.2

6.4.3.1 [1030] Problem 1

problem number 1030

Added Feb. 17, 2019.

Problem Chapter 4.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = (x^2-y^2) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (x^2 - y^2)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {x \left (a^2 \left (x^2-3 y^2\right )+3 a b x y-b^2 x^2\right )}{3 a^3}\right ) c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x) +b*diff(w(x,y),y) = (x^2-y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) {\mathrm e}^{\frac {\left (\left (x^{2}-3 y^{2}\right ) a^{2}+3 a b x y -b^{2} x^{2}\right ) x}{3 a^{3}}}\]

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6.4.3.2 [1031] Problem 2

problem number 1031

Added Feb. 17, 2019.

Problem Chapter 4.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^2 w_x + a x y w_y = b y^2 w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x^2*D[w[x, y], x] + a*x*y*D[w[x, y], y] == b*y^2*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{-\frac {b y^2}{x-2 a x}} c_1\left (y x^{-a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  x^2*diff(w(x,y),x) +a*x*y*diff(w(x,y),y) = b*y^2*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (y \,x^{-a}\right ) {\mathrm e}^{\frac {b \,y^{2}}{x \left (-1+2 a \right )}}\]

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6.4.3.3 [1032] Problem 3

problem number 1032

Added Feb. 17, 2019.

Problem Chapter 4.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^2 w_x + b y^2 w_y = (x+c y) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^2*D[w[x, y], x] + b*y^2*D[w[x, y], y] == (x + c*y)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b}{a x}-\frac {1}{y}\right ) \exp \left (\int _1^x\frac {b y (x-K[1])+a x (c y+K[1])}{a K[1] (b y (x-K[1])+a x K[1])}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde := a*x^2*diff(w(x,y),x) +b*y^2*diff(w(x,y),y) = (x+c*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {a x -y b}{y a x}\right ) \left (\frac {a x}{y}\right )^{-\frac {c}{b}} x^{\frac {c a +b}{b a}}\]

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6.4.3.4 [1033] Problem 4

problem number 1033

Added Feb. 17, 2019.

Problem Chapter 4.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^2 w_x + a y^2 w_y = (b x^2+c x y+d y^2) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x^2*D[w[x, y], x] + a*y^2*D[w[x, y], y] == (b*x^2 + c*x*y + d*y^2)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a}{x}-\frac {1}{y}\right ) \exp \left (\int _1^x\frac {\left (d y^2+K[1] (c y+b K[1])\right ) x^2+a y (x-K[1]) (c y+2 b K[1]) x+a^2 b y^2 (x-K[1])^2}{(a y (x-K[1])+x K[1])^2}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=x^2*diff(w(x,y),x) +a*y^2*diff(w(x,y),y) = (b*x^2+c*x*y+d*y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-y a +x}{y x}\right ) \left (\frac {x}{y}\right )^{-\frac {c y x}{y a -x}} {\mathrm e}^{\frac {x \left (y a -x \right ) b +d \,y^{2}}{y a -x}}\]

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6.4.3.5 [1034] Problem 5

problem number 1034

Added Feb. 17, 2019.

Problem Chapter 4.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ y^2 w_x + a x^2 w_y = (b x^2+c y^2) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  y^2*D[w[x, y], x] + a*x^2*D[w[x, y], y] == (b*x^2 + c*y^2)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\begin{align*}& \left \{w(x,y)\to c_1\left (\frac {1}{3} \left (y^3-a x^3\right )\right ) e^{\frac {b \sqrt [3]{y^3}}{a}+c x}\right \}\\& \left \{w(x,y)\to c_1\left (\frac {1}{3} \left (y^3-a x^3\right )\right ) e^{c x-\frac {\sqrt [3]{-1} b \sqrt [3]{y^3}}{a}}\right \}\\& \left \{w(x,y)\to c_1\left (\frac {1}{3} \left (y^3-a x^3\right )\right ) e^{\frac {(-1)^{2/3} b \sqrt [3]{y^3}}{a}+c x}\right \}\\\end{align*}

Maple ✓

restart; 
pde :=y^2*diff(w(x,y),x) +a*x^2*diff(w(x,y),y) =(b*x^2+c*y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-a \,x^{3}+y^{3}\right ) {\mathrm e}^{x c +\frac {b y}{a}}\]

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6.4.3.6 [1035] Problem 6

problem number 1035

Added Feb. 17, 2019.

Problem Chapter 4.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x y w_x + a y^2 w_y = (b x+c y + d) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*y*D[w[x, y], x] + a*y^2*D[w[x, y], y] == (b*x + c*y + d)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to x^c c_1\left (y x^{-a}\right ) e^{-\frac {\frac {b x}{a-1}+\frac {d}{a}}{y}}\right \}\right \}\]

Maple ✓

restart; 
pde :=x*y*diff(w(x,y),x) +a*y^2*diff(w(x,y),y) =(b*x+c*y+d)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = x^{c} f_{1} \left (y \,x^{-a}\right ) {\mathrm e}^{\frac {\left (-b x -d \right ) a +d}{y \left (-1+a \right ) a}}\]

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6.4.3.7 [1036] Problem 7

problem number 1036

Added Feb. 17, 2019.

Problem Chapter 4.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a y+b) w_x + (a y^2-b x) w_y = a y w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*(a*y + b)*D[w[x, y], x] + (a*y^2 - b*x)*D[w[x, y], y] == a*y*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=x*(a*y+b)*diff(w(x,y),x) +(a*y^2-b*x)*diff(w(x,y),y) =a*y*w(x,y); 
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 {-a \left (x +y \right ) \ln \left (-\frac {a \left (x +y \right ) \left (x a -b \right )}{x \left (a y +b \right )}\right )+a \left (x +y \right ) \ln \left (\frac {-x a +b}{a y +b}\right )-a \left (x +y \right ) \ln \left (2\right )+a y +b}{3 a \left (x +y \right )}\right ) {\mathrm e}^{\frac {\int _{}^{x}\frac {2 \,{\mathrm e}^{\operatorname {RootOf}\left (2 \,{\mathrm e}^{\textit {\_Z}} x a \ln \left (-\frac {a \left (x +y \right ) \left (x a -b \right )}{x \left (a y +b \right )}\right )+2 \,{\mathrm e}^{\textit {\_Z}} a y \ln \left (-\frac {a \left (x +y \right ) \left (x a -b \right )}{x \left (a y +b \right )}\right )-2 \,{\mathrm e}^{\textit {\_Z}} x a \ln \left (-\frac {x a -b}{a y +b}\right )-2 \,{\mathrm e}^{\textit {\_Z}} a y \ln \left (-\frac {x a -b}{a y +b}\right )+2 \,{\mathrm e}^{\textit {\_Z}} x a \ln \left (2\right )+2 \,{\mathrm e}^{\textit {\_Z}} a y \ln \left (2\right )-2 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -b \right )}{\textit {\_a}}\right ) {\mathrm e}^{\textit {\_Z}} a x -2 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -b \right )}{\textit {\_a}}\right ) {\mathrm e}^{\textit {\_Z}} a y +2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x +2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a y -9 x a \ln \left (-\frac {a \left (x +y \right ) \left (x a -b \right )}{x \left (a y +b \right )}\right )-9 a y \ln \left (-\frac {a \left (x +y \right ) \left (x a -b \right )}{x \left (a y +b \right )}\right )+9 x a \ln \left (-\frac {x a -b}{a y +b}\right )+9 a y \ln \left (-\frac {x a -b}{a y +b}\right )-9 x a \ln \left (2\right )-9 a y \ln \left (2\right )+9 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -b \right )}{\textit {\_a}}\right ) a x +9 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -b \right )}{\textit {\_a}}\right ) a y -2 \,{\mathrm e}^{\textit {\_Z}} a y -9 \textit {\_Z} a x -9 \textit {\_Z} a y -2 b \,{\mathrm e}^{\textit {\_Z}}-9 x a +9 b \right )} b +9 \textit {\_a} a -9 b}{\textit {\_a} \left (\textit {\_a} a -b \right )}d \textit {\_a}}{9}}\]

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6.4.3.8 [1037] Problem 8

problem number 1037

Added Feb. 17, 2019.

Problem Chapter 4.2.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(k y-x+a) w_x - y(k x-y +a) w_y = b(y-x) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*(k*y - x + a)*D[w[x, y], x] - y*(k*x - y + a)*D[w[x, y], y] == b*(y - x)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=x*(k*y-x+a)*diff(w(x,y),x)-y*(k*x-y+a)*diff(w(x,y),y) = b*(y-x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime')); 
sol:=simplify(sol);
 

\[\text {Expression too large to display}\]

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