101 HFOPDE, chapter 4.2.3

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101.1 Problem 1

problem number 847

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac { \left ( ya-bx \right ) ^{3}dx}{{a}^{4}}}+3/2\,{\frac { \left ( ya-bx \right ) ^{2}bd{x}^{2}}{{a}^{4}}}+{\frac { \left ( ya-bx \right ) {b}^{2}d{x}^{3}}{{a}^{4}}}+1/4\,{\frac {{x}^{4}c}{a}}+1/4\,{\frac {{b}^{3}d{x}^{4}}{{a}^{4}}}}} \]

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101.2 Problem 2

problem number 848

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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101.3 Problem 3

problem number 849

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) {{\rm e}^{1/2\,{\frac {b{y}^{3}}{x}}+1/2\,{y}^{2}a}} \]

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101.4 Problem 4

problem number 850

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-a} \right ) {x}^{b}{{\rm e}^{-{\frac {ad}{x \left ( a+1 \right ) }}-{\frac {d}{x \left ( a+1 \right ) }}-{\frac {c}{y \left ( a+1 \right ) }}-{\frac {k}{y \left ( a+1 \right ) x}}-{\frac {c}{ \left ( a+1 \right ) ay}}}} \]

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101.5 Problem 5

problem number 851

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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101.6 Problem 6

problem number 852

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y^2-x^2}{2 x^2 y^2}\right ) \exp \left (b x-\frac {c \tan ^{-1}\left (\frac {x \sqrt {\frac {a y^2-x^2}{x^2 y^2}}}{\sqrt {a-\frac {a y^2-x^2}{y^2}}}\right )}{\sqrt {\frac {a y^2-x^2}{x^2 y^2}}}\right )\right \},\left \{w(x,y)\to c_1\left (\frac {a y^2-x^2}{2 x^2 y^2}\right ) \exp \left (\frac {c \tan ^{-1}\left (\frac {x \sqrt {\frac {a y^2-x^2}{x^2 y^2}}}{\sqrt {a-\frac {a y^2-x^2}{y^2}}}\right )}{\sqrt {\frac {a y^2-x^2}{x^2 y^2}}}+b x\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{y}^{2}a-{x}^{2}}{{x}^{2}{y}^{2}}} \right ) {{\rm e}^{bx}} \left ( \sqrt {-{\frac {{y}^{2}a-{x}^{2}}{{x}^{2}{y}^{2}}}}x+\sqrt {-{\frac {{y}^{2}a-{x}^{2}}{{y}^{2}}}+a} \right ) ^{{c{\frac {1}{\sqrt {-{\frac {{y}^{2}a-{x}^{2}}{{x}^{2}{y}^{2}}}}}}}} \]