57 HFOPDE, chapter 2.7.4

 57.1 problem number 1
 57.2 problem number 2
 57.3 problem number 3
 57.4 problem number 4
 57.5 problem number 5
 57.6 problem number 6
 57.7 problem number 7
 57.8 problem number 8
 57.9 problem number 9
 57.10 problem number 10
 57.11 problem number 11
 57.12 problem number 12

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57.1 problem number 1

problem number 532

Added Feb. 1, 2019.

Problem 2.7.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx-\int \!\lambda \, \left ( \pi /2-\arctan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+y \right ) \]

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57.2 problem number 2

problem number 533

Added Feb. 1, 2019.

Problem 2.7.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \lambda \, \left ( \pi /2-\arctan \left ( \lambda \,y \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]

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57.3 problem number 3

problem number 534

Added Feb. 1, 2019.

Problem 2.7.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + k \arccot ^n(a x+b y+c) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac{ax+by}{b}}}\! \left ( k \left ( \pi /2-\arctan \left ({\it \_a}\,b+c \right ) \right ) ^{n}b+a \right ) ^{-1}{d{\it \_a}}b+x \right ) \]

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57.4 problem number 4

problem number 535

Added Feb. 1, 2019.

Problem 2.7.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + k \arccot ^k(\lambda x) \arccot ^n(\mu y) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \pi /2-\arctan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac{ \left ( \pi /2-\arctan \left ( \lambda \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \]

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57.5 problem number 5

problem number 536

Added Feb. 1, 2019.

Problem 2.7.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+ \lambda (\arccot x)^n y - a^2 +a \lambda (\arccot x)^n \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{y\int \!{{\rm e}^{\lambda \,\int \! \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x-2\,ax}}\,{\rm d}x+\int \!{{\rm e}^{\lambda \,\int \! \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x-2\,ax}}\,{\rm d}xa+{{\rm e}^{\lambda \,\int \! \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x-2\,ax}}}{a+y}} \right ) \]

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57.6 problem number 6

problem number 537

Added Feb. 1, 2019.

Problem 2.7.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+ \lambda x (\arccot x)^n y + \lambda (\arccot x)^n \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac{ \left ({\rm arccot} \left (x\right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{ \left ({\rm arccot} \left (x\right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac{ \left ({\rm arccot} \left (x\right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]

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57.7 problem number 7

problem number 538

Added Feb. 1, 2019.

Problem 2.7.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x - \left ( (k+1) x^k y^2- \lambda (\arccot x)^n (x^{k+1} y -1) \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{{x}^{k+1}y-1} \left ( y{x}^{k+1}\int \!{\frac{{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk+y{x}^{k+1}\int \!{\frac{{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x-{{\rm e}^{\int \!{\frac{ \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}{x}^{k+1}\lambda \,x-2\,k-2}{x}}\,{\rm d}x}}{x}^{k+1}-\int \!{\frac{{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk-\int \!{\frac{{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \right ) } \right ) \]

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57.8 problem number 8

problem number 539

Added Feb. 1, 2019.

Problem 2.7.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n y^2+a y + a b -b^2 \lambda (\arccot x)^n n \right ) w_y = 0 \]

Mathematica

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

Maple

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

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57.9 problem number 9

problem number 540

Added Feb. 1, 2019.

Problem 2.7.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n y^2- b \lambda x^m(\arccot x)^n y+ b m x^{m-1} \right ) w_y = 0 \]

Mathematica

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

Maple

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

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57.10 problem number 10

problem number 541

Added Feb. 1, 2019.

Problem 2.7.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n y^2+ b m x^{m-1} - \lambda b^2 x^{2 m} (\arccot x^n) \right ) w_y = 0 \]

Mathematica

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

Maple

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

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57.11 problem number 11

problem number 542

Added Feb. 1, 2019.

Problem 2.7.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n(y-a x^m-b)^2+a m x^{m-1} \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{x}^{m}\int \! \left ({\rm arccot} \left (x\right ) \right ) ^{n}\lambda \,{\rm d}xa-y\int \! \left ({\rm arccot} \left (x\right ) \right ) ^{n}\lambda \,{\rm d}x+\int \! \left ({\rm arccot} \left (x\right ) \right ) ^{n}\lambda \,{\rm d}xb-1}{a{x}^{m}+b-y}} \right ) \]

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57.12 problem number 12

problem number 543

Added Feb. 1, 2019.

Problem 2.7.4.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( \lambda (\arccot x)^n y^2+ k y+ \lambda b^2 x^{2 k} (\arccot x)^n \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \lambda \,b\int \! \left ( \pi /2-\arctan \left ( x \right ) \right ) ^{n}{x}^{k-1}\,{\rm d}x-\arctan \left ({\frac{{x}^{-k}y}{b}} \right ) \right ) \]