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71 HFOPDE, chapter 3.2.1

71.1 Problem 1
71.2 Problem 2
71.3 Problem 3
71.4 Problem 4
71.5 Problem 5
71.6 Problem 6
71.7 Problem 7
71.8 Problem 8

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71.1 Problem 1

problem number 654

Added Feb. 9, 2019.

Problem Chapter 3.2.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

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

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

\[ w \left ( x,y \right ) ={\frac {cx}{a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

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71.2 Problem 2

problem number 655

Added Feb. 9, 2019.

Problem Chapter 3.2.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == alpha*x + beta*y + gamma; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

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

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

\[ w \left ( x,y \right ) =1/2\,{\frac { \left ( a\alpha -b\beta \right ) {x}^{2}}{{a}^{2}}}+ \left ( {\frac {\beta \,y}{a}}+{\frac {\gamma }{a}} \right ) x+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

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71.3 Problem 3

problem number 656

Added Feb. 9, 2019.

Problem Chapter 3.2.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*x*D[w[x, y], x] + b*D[w[x, y], y] == alpha*x + beta*y + gamma; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

\[ \left \{\left \{w(x,y)\to \frac {2 a^2 c_1\left (\frac {a y-b \log (x)}{a}\right )+2 a \alpha x+2 a \beta y \log (x)+2 a \gamma \log (x)-b \beta \log ^2(x)}{2 a^2}\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\frac {\alpha \,x}{a}}+{\frac {\ln \left ( x \right ) \beta \,y}{a}}-1/2\,{\frac {1}{{a}^{2}} \left ( b\beta \, \left ( \ln \left ( x \right ) \right ) ^{2}-2\,\gamma \,\ln \left ( x \right ) a-2\,{\it \_F1} \left ( -{\frac {b\ln \left ( x \right ) -ya}{a}} \right ) {a}^{2} \right ) } \]

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71.4 Problem 4

problem number 657

Added Feb. 9, 2019.

Problem Chapter 3.2.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b x w_y = c \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*x*D[w[x, y], x] + b*x*D[w[x, y], y] == c; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

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

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

\[ w \left ( x,y \right ) ={\frac {1}{a} \left ( c\ln \left ( x \right ) +{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) a \right ) } \]

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71.5 Problem 5

problem number 658

Added Feb. 9, 2019.

Problem Chapter 3.2.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x +b) w_x + (c y +d) w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*x + beta*y + gamma; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

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

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

\[ w \left ( x,y \right ) ={\frac {\alpha \,x}{a}}+{\frac {\beta \,y}{c}}+{\frac {1}{{a}^{2}{c}^{2}} \left ( {\it \_F1} \left ( {\frac {cy+d}{c} \left ( ax+b \right ) ^{-{\frac {c}{a}}}} \right ) {a}^{2}{c}^{2}+\ln \left ( ax+b \right ) \gamma \,a{c}^{2}-\ln \left ( ax+b \right ) \beta \,dac-\ln \left ( ax+b \right ) \alpha \,b{c}^{2}+{a}^{2}\beta \,d \right ) } \]

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71.6 Problem 6

problem number 659

Added Feb. 9, 2019.

Problem Chapter 3.2.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*y*D[w[x, y], x] + b*D[w[x, y], y] == alpha*x + beta*y + gamma; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

\[ \left \{\left \{w(x,y)\to \frac {-a^{3/2} \alpha y^2 \sqrt {a y^2}+3 \sqrt {a} \alpha b x \sqrt {a y^2}+3 a b^2 c_1\left (\frac {a y^2-2 b x}{2 a}\right )+3 \sqrt {a} b \gamma \sqrt {a y^2}+3 b^2 \beta x}{3 a b^2}\right \},\left \{w(x,y)\to \frac {a^{3/2} \alpha y^2 \sqrt {a y^2}-3 \sqrt {a} \alpha b x \sqrt {a y^2}+3 a b^2 c_1\left (\frac {a y^2-2 b x}{2 a}\right )-3 \sqrt {a} b \gamma \sqrt {a y^2}+3 b^2 \beta x}{3 a b^2}\right \}\right \} \]

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

\[ w \left ( x,y \right ) =1/6\,{\frac { \left ( 6\,\sqrt { \left ( {y}^{2}a-2\,bx \right ) a+2\,axb}a\alpha \,b+6\,a{b}^{2}\beta \right ) x}{{b}^{2}{a}^{2}}}-1/2\,{\frac {\sqrt { \left ( {y}^{2}a-2\,bx \right ) a+2\,axb}\alpha \,{y}^{2}}{{b}^{2}}}+1/6\,{\frac {1}{{b}^{2}{a}^{2}} \left ( 6\,{\it \_F1} \left ( {\frac {{y}^{2}a-2\,bx}{a}} \right ) {b}^{2}{a}^{2}+ \left ( \left ( {y}^{2}a-2\,bx \right ) a+2\,axb \right ) ^{3/2}\alpha +6\,\gamma \,\sqrt { \left ( {y}^{2}a-2\,bx \right ) a+2\,axb}ba \right ) } \]

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71.7 Problem 7

problem number 660

Added Feb. 9, 2019.

Problem Chapter 3.2.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b x w_y = c \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == c; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

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

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

\[ w \left ( x,y \right ) ={\frac {1}{\sqrt {ab}} \left ( {\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \sqrt {ab}+c\ln \left ( {\frac {axb}{\sqrt {ab}}}+\sqrt {ab{x}^{2}+ \left ( {y}^{2}a-b{x}^{2} \right ) a} \right ) \right ) } \]

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71.8 Problem 8

problem number 661

Added Feb. 9, 2019.

Problem Chapter 3.2.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b x w_y = c x+ k y \]

Mathematica ✓ 

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == c*x + k*y; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];

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

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

\[ w \left ( x,y \right ) ={\frac {cy}{b}}+{\frac {kx}{a}}+{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \]

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