96 HFOPDE, chapter 4.2.1

 96.1 Problem 1
 96.2 Problem 2
 96.3 Problem 3
 96.4 Problem 4
 96.5 Problem 5
 96.6 Problem 6
 96.7 Problem 7

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96.1 Problem 1

problem number 816

Added Feb. 17, 2019.

Problem Chapter 4.2.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){{\rm e}^{{\frac{cx}{a}}}} \]

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96.2 Problem 2

problem number 817

Added Feb. 17, 2019.

Problem Chapter 4.2.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + y w_y = b w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-{\frac{x}{a}}}} \right ){{\rm e}^{{\frac{bx}{a}}}} \]

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96.3 Problem 3

problem number 818

Added Feb. 17, 2019.

Problem Chapter 4.2.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a w \]

Mathematica

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

Maple

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

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96.4 Problem 4

problem number 819

Added Feb. 17, 2019.

Problem Chapter 4.2.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x ( a w_x - b w_y ) = c y w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( bx+y \right ){x}^{c \left ( bx+y \right ) }{{\rm e}^{-bcx}} \]

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96.5 Problem 5

problem number 820

Added Feb. 17, 2019.

Problem Chapter 4.2.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={{\rm e}^{ax}}{\it \_F1} \left ({\frac{y}{x}} \right ) \]

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96.6 Problem 6

problem number 821

Added Feb. 17, 2019.

Problem Chapter 4.2.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (x-a) w_x + (y-b) w_y = w \]

Mathematica

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

Maple

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

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96.7 Problem 7

problem number 822

Added Feb. 17, 2019.

Problem Chapter 4.2.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (y+a x) w_x + (y- a x) w_y = b w \]

Mathematica

\[ \text{DSolve}\left [(y-a x) w^{(0,1)}(x,y)+(a x+y) w^{(1,0)}(x,y)=b w(x,y),w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]