59 HFOPDE, chapter 2.8.2

 59.1 problem number 1
 59.2 problem number 2
 59.3 problem number 3
 59.4 problem number 4
 59.5 problem number 5
 59.6 problem number 6
 59.7 problem number 7
 59.8 problem number 8
 59.9 problem number 9
 59.10 problem number 10
 59.11 problem number 11

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59.1 problem number 1

problem number 557

Added Feb. 4, 2019.

Problem 2.8.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a e^{\lambda x} y^2 + a e^{\lambda x} f(x) y+\lambda f(x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y f(x) e^{\lambda x}+a y^2 e^{\lambda x}+\lambda f(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{\lambda \,{{\rm e}^{\lambda \,x}} \left ( a{{\rm e}^{\lambda \,x}}y+\lambda \right ) }{\int \!{{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x-\lambda \,x}}\,{\rm d}x{{\rm e}^{2\,\lambda \,x}}ya+\lambda \,\int \!{{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x-\lambda \,x}}\,{\rm d}x{{\rm e}^{\lambda \,x}}+{{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x}}}} \right ) \]

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59.2 problem number 2

problem number 558

Added Feb. 4, 2019.

Problem 2.8.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2-a e^{\lambda x} f(x) y+a \lambda e^{\lambda x}\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a y f(x) e^{\lambda x}+a \lambda e^{\lambda x}+y^2 f(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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59.3 problem number 3

problem number 559

Added Feb. 4, 2019.

Problem 2.8.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+a \lambda e^{\lambda x}-a^2 e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a^2 f(x) \left (-e^{2 \lambda x}\right )+a \lambda e^{\lambda x}+y^2 f(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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59.4 problem number 4

problem number 560

Added Feb. 4, 2019.

Problem 2.8.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+\lambda y+ a e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{y e^{-\lambda x}}{\sqrt{a}}\right )-\sqrt{a} \int _1^x f(K[1]) e^{\lambda K[1]} \, dK[1]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \sqrt{a}\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x-\arctan \left ({\frac{{{\rm e}^{-\lambda \,x}}y}{\sqrt{a}}} \right ) \right ) \]

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59.5 problem number 5

problem number 561

Added Feb. 4, 2019.

Problem 2.8.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2-(a e^{\lambda x}+b) f(x) y+a \lambda e^{\lambda x}\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-y f(x) \left (a e^{\lambda x}+b\right )+a \lambda e^{\lambda x}+y^2 f(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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59.6 problem number 6

problem number 562

Added Feb. 4, 2019.

Problem 2.8.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (e^{\lambda x} f(x) y^2+(a f(x)-\lambda ) y+b e^{-\lambda x} f(x)\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (y (a f(x)-\lambda )+b f(x) e^{-\lambda x}+y^2 f(x) e^{\lambda x}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{a}{\sqrt{{a}^{2} \left ({a}^{2}-4\,b \right ) }} \left ( -\sqrt{{a}^{2} \left ({a}^{2}-4\,b \right ) }\int \!f \left ( x \right ) \,{\rm d}x-2\,a\arctanh \left ({\frac{a \left ( 2\,{{\rm e}^{\lambda \,x}}y+a \right ) }{\sqrt{{a}^{2} \left ({a}^{2}-4\,b \right ) }}} \right ) \right ) } \right ) \]

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59.7 problem number 7

problem number 563

Added Feb. 4, 2019.

Problem 2.8.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+ g(x) y+a \lambda e^{\lambda x} -a e^{\lambda x} g(x) -a^2 e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a^2 f(x) \left (-e^{2 \lambda x}\right )-a g(x) e^{\lambda x}+a \lambda e^{\lambda x}+y^2 f(x)+y g(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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59.8 problem number 8

problem number 564

Added Feb. 7, 2019.

Problem 2.8.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2- a e^{\lambda x} g(x) y + a \lambda e^{\lambda x} +a^2 e^{2 \lambda x} (g(x)-f(x))\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a^2 e^{2 \lambda x} (g(x)-f(x))-a y g(x) e^{\lambda x}+a \lambda e^{\lambda x}+y^2 f(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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59.9 problem number 9

problem number 565

Added Feb. 7, 2019.

Problem 2.8.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+2 a \lambda x e^{\lambda x^2} - a^2 f(x) e^{2 \lambda x^2} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a^2 f(x) \left (-e^{2 \lambda x^2}\right )+2 a \lambda x e^{\lambda x^2}+y^2 f(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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59.10 problem number 10

problem number 566

Added Feb. 7, 2019.

Problem 2.8.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+2 \lambda x y+ a f(x) e^{2 \lambda x^2} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{y e^{-\lambda x^2}}{\sqrt{a}}\right )-\sqrt{a} \int _1^x f(K[1]) e^{\lambda K[1]^2} \, dK[1]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \sqrt{a}\int \!{{\rm e}^{{x}^{2}\lambda }}f \left ( x \right ) \,{\rm d}x-\arctan \left ({\frac{{{\rm e}^{-{x}^{2}\lambda }}y}{\sqrt{a}}} \right ) \right ) \]

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59.11 problem number 11

problem number 567

Added Feb. 7, 2019.

Problem 2.8.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) e^{\lambda y} + g(x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (f(x) e^{\lambda y}+g(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{\lambda \,\int \!f \left ( x \right ){{\rm e}^{\lambda \,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\lambda \, \left ( \int \!g \left ( x \right ) \,{\rm d}x-y \right ) }}}{\lambda }} \right ) \]