6.3.3 2.2

6.3.3.1 [853] Problem 1

problem number 853

Added Feb. 9, 2019.

Problem Chapter 3.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c x^2+d y^2+ k x y+n \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{6\,{a}^{3}} \left ( 6\,{\it \_F1} \left ( {\frac {ay-bx}{a}} \right ) {a}^{3}+2\, \left ( \left ( c{x}^{2}+3\,d{y}^{2}+3/2\,kxy+3\,n \right ) {a}^{2}-3\, \left ( dy+1/6\,kx \right ) bxa+{b}^{2}d{x}^{2} \right ) x \right ) }\]

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6.3.3.2 [854] Problem 2

problem number 854

Added Feb. 9, 2019.

Problem Chapter 3.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c x^2+d y^2+ k x y+n \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {kxy}{a+b}}+{\frac {c{x}^{2}}{2\,a}}+{\frac {n\ln \left ( x \right ) }{a}}+{\frac {d{y}^{2}}{2\,b}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]

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6.3.3.3 [855] Problem 3

problem number 855

Added Feb. 9, 2019.

Problem Chapter 3.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

Maple ✓

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

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

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6.3.3.4 [856] Problem 4

problem number 856

Added Feb. 9, 2019.

Problem Chapter 3.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^2 w_x + b y^2 w_y = c x^2+d y^2+ k x y+ n x+ m y+s \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {a b x (a x-b y) c_1\left (\frac {b}{a x}-\frac {1}{y}\right )-a^2 m x^2 \log \left (\frac {a x}{y}\right )+a b c x^3-a b d x y^2+a b k x^2 y \log \left (\frac {a x}{y}\right )+x \log (x) (a m+b n) (a x-b y)+a b m x y \log \left (\frac {a x}{y}\right )-a b s x-b^2 c x^2 y+b^2 s y}{a b x (a x-b y)}\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {xc}{a}}-{\frac {d{y}^{2}}{ax-by}}-{\frac {m}{b}\ln \left ( {\frac {ax}{y}} \right ) }+{\frac {kxy}{ax-by}\ln \left ( {\frac {ax}{y}} \right ) }-{\frac {s}{ax}}+{\frac {m\ln \left ( x \right ) }{b}}+{\frac {n\ln \left ( x \right ) }{a}}+{\it \_F1} \left ( {\frac {ax-by}{ayx}} \right ) \]

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6.3.3.5 [857] Problem 5

problem number 857

Added Feb. 9, 2019.

Problem Chapter 3.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + a x y w_y = b y^2 \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.3.6 [858] Problem 6

problem number 858

Added Feb. 9, 2019.

Problem Chapter 3.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^2 w_x + b x^2 w_y = c x^2+d \]

Mathematica ✓

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

\begin {align*} & \left \{w(x,y)\to \frac {b d x \left (\frac {a y^3}{a y^3-b x^3}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {b x^3}{b x^3-a y^3}\right )+a c y^3}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}+c_1\left (\frac {a y^3-b x^3}{3 a}\right )\right \}\\& \left \{w(x,y)\to -\frac {\sqrt [3]{-1} \left (b d x \left (\frac {a y^3}{a y^3-b x^3}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {b x^3}{b x^3-a y^3}\right )+a c y^3\right )}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}+c_1\left (\frac {a y^3-b x^3}{3 a}\right )\right \}\\& \left \{w(x,y)\to \frac {(-1)^{2/3} \left (b d x \left (\frac {a y^3}{a y^3-b x^3}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {b x^3}{b x^3-a y^3}\right )+a c y^3\right )}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}+c_1\left (\frac {a y^3-b x^3}{3 a}\right )\right \}\\ \end {align*}

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!{ \left ( {{\it \_a}}^{2}c+d \right ) a \left ( \left ( {{\it \_a}}^{3}b+\RootOf \left ( ay-\sqrt [3]{{a}^{2}b{x}^{3}+{a}^{3}{\it \_Z}} \right ) a \right ) {a}^{2} \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_F1} \left ( \RootOf \left ( ay-\sqrt [3]{{a}^{2}b{x}^{3}+{a}^{3}{\it \_Z}} \right ) \right ) \] Contains unresolved integral with RootOf

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6.3.3.7 [859] Problem 7

problem number 859

Added Feb. 9, 2019.

Problem Chapter 3.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^2 w_x + b x y w_y = c x^2+d y^2 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right )-\frac {\sqrt {a} c \sqrt {y^2-\frac {b x^2}{a}} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {y^2-\frac {b x^2}{a}}}\right )}{b^{3/2}}+\frac {d x}{a}+\frac {c x}{b}\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{ab} \left ( -ca \left ( {y}^{2}a-{x}^{2}b \right ) \arctan \left ( {bx{\frac {1}{\sqrt {b \left ( {y}^{2}a-{x}^{2}b \right ) }}}} \right ) +\sqrt {b \left ( {y}^{2}a-{x}^{2}b \right ) } \left ( {\it \_F1} \left ( {\frac {{y}^{2}a-{x}^{2}b}{a}} \right ) ab+ \left ( ac+bd \right ) x \right ) \right ) {\frac {1}{\sqrt {b \left ( {y}^{2}a-{x}^{2}b \right ) }}}}\]

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