6.3.4 2.3

6.3.4.1 [860] Problem 1

problem number 860

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'));
 

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

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6.3.4.2 [861] Problem 2

problem number 861

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'));
 

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

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6.3.4.3 [862] Problem 3

problem number 862

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 \text {arctanh}\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 \text {arctanh}\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'));
 

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

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6.3.4.4 [863] Problem 4

problem number 863

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )+\frac {b^2 k x}{a^3}-\frac {b k x^2}{2 a^2}-\frac {\log (a x+b) \left (a^3 d^3 n+b^3 c^3 k\right )}{a^4 c^3}+\frac {k x^3}{3 a}+\frac {n \left (2 c^3 y^3-3 c^2 d y^2+6 c d^2 y+11 d^3\right )}{6 c^4}\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {6 f_{1} \left (\frac {\left (c y +d \right ) \left (a x +b \right )^{-\frac {c}{a}}}{c}\right ) a^{4} c^{4}+\left (-6 a^{3} c \,d^{3} n -6 b^{3} c^{4} k \right ) \ln \left (a x +b \right )+2 a \left (\left (c^{2} y^{2}-\frac {5}{2} c d y +\frac {11}{2} d^{2}\right ) \left (c y +d \right ) n \,a^{3}+a^{2} c^{4} k \,x^{3}-\frac {3 a b \,c^{4} k \,x^{2}}{2}+3 b^{2} c^{4} k x \right )}{6 a^{4} c^{4}}\]

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6.3.4.5 [864] Problem 5

problem number 864

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'));
 

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

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6.3.4.6 [865] Problem 6

problem number 865

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'));
 

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

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