80 HFOPDE, chapter 3.5.2

 80.1 Problem 1
 80.2 Problem 2
 80.3 Problem 3

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80.1 Problem 1

problem number 723

Added Feb. 11, 2019.

Problem Chapter 3.5.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c x^n+ s \ln ^k(\lambda y) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{y}{a \left ( n+1 \right ) b} \left ( \ln \left ({\frac{x\lambda \,b}{a}}+{\frac{\lambda \, \left ( ay-bx \right ) }{a}} \right ) ans+\ln \left ({\frac{x\lambda \,b}{a}}+{\frac{\lambda \, \left ( ay-bx \right ) }{a}} \right ) as-ans-as \right ) }+{\frac{1}{a \left ( n+1 \right ) b} \left ({\it \_F1} \left ({\frac{ay-bx}{a}} \right ) abn+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) ba+{x}^{n+1}cb \right ) } \]

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80.2 Problem 2

problem number 724

Added Feb. 11, 2019.

Problem Chapter 3.5.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = b y^2+c x^n y+ s \ln ^k(\lambda x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{(-\log (\lambda x))^{-k} \left (3 n^2 s \log ^k(\lambda x) \text{Gamma}(k+1,-\log (\lambda x))+9 n s \log ^k(\lambda x) \text{Gamma}(k+1,-\log (\lambda x))+6 s \log ^k(\lambda x) \text{Gamma}(k+1,-\log (\lambda x))+a^2 b \lambda n^2 x^3 (-\log (\lambda x))^k+3 a^2 b \lambda n x^3 (-\log (\lambda x))^k+2 a^2 b \lambda x^3 (-\log (\lambda x))^k-3 a b \lambda n^2 x^2 y (-\log (\lambda x))^k-9 a b \lambda n x^2 y (-\log (\lambda x))^k-6 a b \lambda x^2 y (-\log (\lambda x))^k+3 \lambda n^2 c_1(y-a x) (-\log (\lambda x))^k-3 a c \lambda x^{n+2} (-\log (\lambda x))^k+9 \lambda n c_1(y-a x) (-\log (\lambda x))^k+6 \lambda c_1(y-a x) (-\log (\lambda x))^k+3 b \lambda n^2 x y^2 (-\log (\lambda x))^k+9 b \lambda n x y^2 (-\log (\lambda x))^k+6 b \lambda x y^2 (-\log (\lambda x))^k+6 c \lambda y x^{n+1} (-\log (\lambda x))^k+3 c \lambda n y x^{n+1} (-\log (\lambda x))^k\right )}{3 \lambda (n+1) (n+2)}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{a}^{2}b{{\it \_a}}^{2}+a{{\it \_a}}^{n+1}c+2\, \left ( -ax+y \right ) ab{\it \_a}+{{\it \_a}}^{n} \left ( -ax+y \right ) c+ \left ( -ax+y \right ) ^{2}b+s \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \] Result has unresolved integrals

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80.3 Problem 3

problem number 725

Added Feb. 11, 2019.

Problem Chapter 3.5.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = b ln^k(\lambda x) \ln ^n(\beta y) \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!b \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k} \left ( \ln \left ( \beta \, \left ({\it \_a}\,a-ax+y \right ) \right ) \right ) ^{n}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]