70 HFOPDE, chapter 3.2.3

 70.1 Problem 1
 70.2 Problem 2
 70.3 Problem 3
 70.4 Problem 4
 70.5 Problem 5
 70.6 Problem 6

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70.1 Problem 1

problem number 653

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

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

Maple

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

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70.2 Problem 2

problem number 654

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

\[ \left \{\left \{w(x,y)\to \frac{2 a^3 c_1\left (y x^{-\frac{b}{a}}\right )+5 a^2 b c_1\left (y x^{-\frac{b}{a}}\right )+2 a^2 c x y^2+a^2 d x^2 y+2 a^2 k \log (x)+2 a b^2 c_1\left (y x^{-\frac{b}{a}}\right )+a b c x y^2+2 a b d x^2 y+5 a b k \log (x)+2 b^2 k \log (x)}{a (2 a+b) (a+2 b)}\right \}\right \} \]

Maple

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

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70.3 Problem 3

problem number 655

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

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

Maple

\[ w \left ( x,y \right ) = \left ( -{\frac{cy}{a}}+{\frac{\sqrt{ab{x}^{2}+ \left ({y}^{2}a-b{x}^{2} \right ) a}c}{{a}^{2}}} \right ){x}^{2}+{\frac{c{y}^{3}}{b}}-2/3\,{\frac{\sqrt{ab{x}^{2}+ \left ({y}^{2}a-b{x}^{2} \right ) a}c{y}^{2}}{ab}}+1/3\,{\frac{1}{\sqrt{ab}{a}^{2}b} \left ( 3\,{\it \_F1} \left ({\frac{{y}^{2}a-b{x}^{2}}{a}} \right ) \sqrt{ab}{a}^{2}b+3\,d\ln \left ({\frac{abx}{\sqrt{ab}}}+\sqrt{ab{x}^{2}+ \left ({y}^{2}a-b{x}^{2} \right ) a} \right ){a}^{2}b \right ) } \]

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70.4 Problem 4

problem number 656

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

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

Maple

\[ w \left ( x,y \right ) =1/3\,{\frac{k{x}^{3}}{a}}-1/2\,{\frac{k{x}^{2}b}{{a}^{2}}}+{\frac{{b}^{2}kx}{{a}^{3}}}+1/3\,{\frac{n{y}^{3}}{c}}-1/2\,{\frac{dn{y}^{2}}{{c}^{2}}}+{\frac{{d}^{2}ny}{{c}^{3}}}+1/6\,{\frac{1}{{a}^{4}{c}^{4}} \left ( 6\,{\it \_F1} \left ({\frac{cy+d}{c} \left ( ax+b \right ) ^{-{\frac{c}{a}}}} \right ){a}^{4}{c}^{4}-6\,\ln \left ( ax+b \right ){d}^{3}n{a}^{3}c-6\,\ln \left ( ax+b \right ){b}^{3}k{c}^{4}+11\,{a}^{4}{d}^{3}n \right ) } \]

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70.5 Problem 5

problem number 657

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

\[ \left \{\left \{w(x,y)\to \frac{a x y^2+b y^3+2 x c_1\left (\frac{y}{x}\right )}{2 x}\right \}\right \} \]

Maple

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

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70.6 Problem 6

problem number 658

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

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

Maple

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