56 HFOPDE, chapter 2.7.3

 56.1 problem number 1
 56.2 problem number 2
 56.3 problem number 3
 56.4 problem number 4
 56.5 problem number 5
 56.6 problem number 6
 56.7 problem number 7
 56.8 problem number 8
 56.9 problem number 9
 56.10 problem number 10
 56.11 problem number 11
 56.12 problem number 12

_______________________________________________________________________________________

56.1 problem number 1

problem number 520

Added January 29, 2019.

Problem 2.7.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Timed out} \]

Maple

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

_______________________________________________________________________________________

56.2 problem number 2

problem number 521

Added January 29, 2019.

Problem 2.7.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.3 problem number 3

problem number 522

Added January 29, 2019.

Problem 2.7.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Timed out} \]

Maple

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

_______________________________________________________________________________________

56.4 problem number 4

problem number 523

Added January 29, 2019.

Problem 2.7.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Timed out} \]

Maple

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

_______________________________________________________________________________________

56.5 problem number 5

problem number 524

Added January 29, 2019.

Problem 2.7.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.6 problem number 6

problem number 525

Added January 29, 2019.

Problem 2.7.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.7 problem number 7

problem number 526

Added Feb. 1, 2019.

Problem 2.7.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.8 problem number 8

problem number 527

Added Feb. 1, 2019.

Problem 2.7.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.9 problem number 9

problem number 528

Added Feb. 1, 2019.

Problem 2.7.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.10 problem number 10

problem number 529

Added Feb. 1, 2019.

Problem 2.7.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.11 problem number 11

problem number 530

Added Feb. 1, 2019.

Problem 2.7.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

56.12 problem number 12

problem number 531

Added Feb. 1, 2019.

Problem 2.7.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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