6.4.4 2.3

6.4.4.1 [1038] Problem 1

problem number 1038

Added Feb. 17, 2019.

Problem Chapter 4.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c x^3+d y^3) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (c*x^3 + d*y^3)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) e^{\frac {1}{4} \left (\frac {c x^4}{a}+\frac {d y^4}{b}\right )}\right \}\right \}\]

Maple ✓

restart; 
pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = (c*x^3+d*y^3)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (c x^{3} a^{3}+4 a^{3} d y^{3}-6 a^{2} b d x y^{2}+4 a b^{2} d x^{2} y -b^{3} d x^{3}\right ) x}{4 a^{4}}}\]

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6.4.4.2 [1039] Problem 2

problem number 1039

Added Feb. 17, 2019.

Problem Chapter 4.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a \sqrt {x^2+y^2} w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {y}{x}\right ) {\mathrm e}^{\sqrt {x^{2}+y^{2}}\, a}\]

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6.4.4.3 [1040] Problem 3

problem number 1040

Added Feb. 17, 2019.

Problem Chapter 4.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )}\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {y}{x}\right ) {\mathrm e}^{\frac {a y^{2}}{2}+\frac {b y^{3}}{2 x}}\]

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6.4.4.4 [1041] Problem 4

problem number 1041

Added Feb. 17, 2019.

Problem Chapter 4.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = x^{b} \mathit {\_F1} \left (y x^{-a}\right ) {\mathrm e}^{\frac {-a^{2} d y -c x +\left (-c x -d y -k \right ) a}{\left (a +1\right ) a x y}}\]

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6.4.4.5 [1042] Problem 5

problem number 1042

Added Feb. 17, 2019.

Problem Chapter 4.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x y^2 w_x + b x^2 y w_y = (a n y^2+ b m x^2) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = x^{n} \left (a y^{2}\right )^{\frac {m}{2}} \mathit {\_F1} \left (\frac {a y^{2}-b x^{2}}{a}\right )\]

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6.4.4.6 [1043] Problem 6

problem number 1043

Added Feb. 17, 2019.

Problem Chapter 4.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x^3 w_x + a y^3 w_y = x^2 (b x + c y) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x^3*D[w[x, y], x] + a*y^3*D[w[x, y], y] == x^2*(b*x + c*y)*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}{2} \left (\frac {a}{x^2}-\frac {1}{y^2}\right )\right ) \exp \left (b x-\frac {c \tan ^{-1}\left (\frac {x \sqrt {\frac {a}{x^2}-\frac {1}{y^2}}}{\sqrt {\frac {x^2}{y^2}}}\right )}{\sqrt {\frac {a}{x^2}-\frac {1}{y^2}}}\right )\right \}\\& \left \{w(x,y)\to c_1\left (\frac {1}{2} \left (\frac {a}{x^2}-\frac {1}{y^2}\right )\right ) \exp \left (\frac {c \tan ^{-1}\left (\frac {x \sqrt {\frac {a}{x^2}-\frac {1}{y^2}}}{\sqrt {\frac {x^2}{y^2}}}\right )}{\sqrt {\frac {a}{x^2}-\frac {1}{y^2}}}+b x\right )\right \}\\ \end {align*}

Maple ✓

restart; 
pde :=x^3*diff(w(x,y),x)+a*y^3*diff(w(x,y),y) =  x^2*(b*x+c*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \left (\sqrt {\frac {-a y^{2}+x^{2}}{x^{2} y^{2}}}\, x +\sqrt {\frac {x^{2}}{y^{2}}}\right )^{\frac {c}{\sqrt {\frac {-a y^{2}+x^{2}}{x^{2} y^{2}}}}} \mathit {\_F1} \left (\frac {-a y^{2}+x^{2}}{x^{2} y^{2}}\right ) {\mathrm e}^{b x}\]

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