47 HFOPDE, chapter 2.5.1

 47.1 problem number 1
 47.2 problem number 3
 47.3 problem number 4

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47.1 problem number 1

problem number 406

Added January 14, 2019.

Problem 2.5.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \ln ^k(\lambda x)+ b\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx+y-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x \right ) \]

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47.2 problem number 3

problem number 407

Added January 14, 2019.

Problem 2.5.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \ln ^k(\lambda y)+ b\right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \frac{1}{a \log ^k(\lambda K[1])+b} \, dK[1]-x\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \ln \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]

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47.3 problem number 4

problem number 408

Added January 14, 2019.

Problem 2.5.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \ln ^k(x+\lambda y)\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac{y\lambda +x}{\lambda }}}\! \left ( 1+a \left ( \ln \left ({\it \_a}\,\lambda \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}\lambda +x \right ) \]