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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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]];
 

\begin{align*}& \left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right ) \exp \left (\int _1^x\frac {a y^2+b m K[1]^2+b \left (K[1]^2-x^2\right )}{K[1] \left (a y^2+b \left (K[1]^2-x^2\right )\right )}dK[1]\right )\right \}\\& \left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right ) \exp \left (\int _1^x\frac {2 a y^2+b m K[2]^2+2 b \left (K[2]^2-x^2\right )}{K[2] \left (a y^2+b \left (K[2]^2-x^2\right )\right )}dK[2]\right )\right \}\\\end{align*}

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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 \sqrt {\frac {x^2}{a y^2}} \text {arcsinh}\left (x \sqrt {\frac {1}{a y^2}-\frac {1}{x^2}}\right )}{\sqrt {\frac {x^2}{y^2}} \sqrt {\frac {1}{a y^2}-\frac {1}{x^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 \sqrt {\frac {x^2}{a y^2}} \text {arcsinh}\left (x \sqrt {\frac {1}{a y^2}-\frac {1}{x^2}}\right )}{\sqrt {\frac {x^2}{y^2}} \sqrt {\frac {1}{a y^2}-\frac {1}{x^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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________