87 HFOPDE, chapter 3.7.1

 87.1 Problem 1
 87.2 Problem 2
 87.3 Problem 3
 87.4 Problem 4
 87.5 Problem 5

_______________________________________________________________________________________

87.1 Problem 1

problem number 755

Added Feb. 11, 2019.

Problem Chapter 3.7.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arcsin \frac{x}{\lambda }+ k \arcsin \frac{y}{\beta } \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{-\frac{i b k x \sqrt{a^2 \left (\beta ^2-y^2\right )} \log \left (2 \left (\sqrt{a^2 \left (\beta ^2-y^2\right )}-i a y\right )\right )}{\sqrt{1-\frac{y^2}{\beta ^2}}}+a^2 b \beta c_1\left (y-\frac{b x}{a}\right )-\frac{a^2 k y^2}{\sqrt{1-\frac{y^2}{\beta ^2}}}+\frac{a^2 \beta ^2 k}{\sqrt{1-\frac{y^2}{\beta ^2}}}+\frac{i a k y \sqrt{a^2 \left (\beta ^2-y^2\right )} \log \left (2 \left (\sqrt{a^2 \left (\beta ^2-y^2\right )}-i a y\right )\right )}{\sqrt{1-\frac{y^2}{\beta ^2}}}+a b \beta c \lambda \sqrt{1-\frac{x^2}{\lambda ^2}}+a b \beta c x \sin ^{-1}\left (\frac{x}{\lambda }\right )+a b \beta k x \sin ^{-1}\left (\frac{y}{\beta }\right )}{a^2 b \beta }\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{1}{ab} \left ( \sqrt{{\frac{{\lambda }^{2}-{x}^{2}}{{\lambda }^{2}}}}bc\lambda +\arcsin \left ({\frac{x}{\lambda }} \right ) bcx+\arcsin \left ({\frac{y}{\beta }} \right ) kya+\sqrt{{\frac{{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}a\beta \,k+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) ba \right ) } \]

_______________________________________________________________________________________

87.2 Problem 2

problem number 756

Added Feb. 11, 2019.

Problem Chapter 3.7.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arcsin (\lambda x+\beta y) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{c \left (\sqrt{-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}+(\beta y+\lambda x) \sin ^{-1}(\beta y+\lambda x)\right )}{a \lambda +b \beta }+c_1\left (y-\frac{b x}{a}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\lambda \,a+b\beta } \left ( \sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}c+ \left ( \lambda \,a+b\beta \right ){\it \_F1} \left ({\frac{ay-bx}{a}} \right ) +c\arcsin \left ( \beta \,y+\lambda \,x \right ) \left ( \beta \,y+\lambda \,x \right ) \right ) } \]

_______________________________________________________________________________________

87.3 Problem 3

problem number 757

Added Feb. 11, 2019.

Problem Chapter 3.7.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \arcsin (\lambda x+\beta y) \]

Mathematica

\[ \left \{\left \{w(x,y)\to a x \left (\frac{\sqrt{-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}}{\beta y+\lambda x}+\sin ^{-1}(\beta y+\lambda x)\right )+c_1\left (\frac{y}{x}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={a \left ( x \left ({\frac{\beta \,y}{x}}+\lambda \right ) \arcsin \left ( x \left ({\frac{\beta \,y}{x}}+\lambda \right ) \right ) +\sqrt{-{x}^{2} \left ({\frac{\beta \,y}{x}}+\lambda \right ) ^{2}+1} \right ) \left ({\frac{\beta \,y}{x}}+\lambda \right ) ^{-1}}+{\it \_F1} \left ({\frac{y}{x}} \right ) \]

_______________________________________________________________________________________

87.4 Problem 4

problem number 758

Added Feb. 11, 2019.

Problem Chapter 3.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arcsin ^n(\lambda x) w_y = c \arcsin ^m(\mu x)+s \arcsin ^k(\beta y) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\! \left ( \arcsin \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+{\frac{1}{a} \left ( \arcsin \left ({\frac{\beta \,\arcsin \left ( \lambda \,{\it \_a} \right ) b{\it \_a}}{a}}-{\frac{ \left ( \arcsin \left ( \lambda \,x \right ) bx\lambda -\lambda \,ya+\sqrt{-{\lambda }^{2}{x}^{2}+1}b \right ) \beta }{\lambda \,a}}+{\frac{\beta \,\sqrt{-{{\it \_a}}^{2}{\lambda }^{2}+1}b}{\lambda \,a}} \right ) \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ({\frac{-\arcsin \left ( \lambda \,x \right ) bx\lambda +\lambda \,ya-\sqrt{-{\lambda }^{2}{x}^{2}+1}b}{\lambda \,a}} \right ) \]

_______________________________________________________________________________________

87.5 Problem 5

problem number 759

Added Feb. 11, 2019.

Problem Chapter 3.7.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arcsin ^n(\lambda y) w_y = c \arcsin ^m(\mu x)+s \arcsin ^k(\beta y) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{y}\!{\frac{a}{b\arcsin \left ( \lambda \,{\it \_a} \right ) } \left ( \arcsin \left ({\frac{\mu \, \left ( x\lambda \,b-a\Ci \left ( \arcsin \left ( \lambda \,y \right ) \right ) \right ) }{\lambda \,b}}+{\frac{\mu \,a\Ci \left ( \arcsin \left ( \lambda \,{\it \_a} \right ) \right ) }{\lambda \,b}} \right ) \right ) ^{m}}+{\frac{ \left ( \arcsin \left ({\it \_a}\,\beta \right ) \right ) ^{k}}{b\arcsin \left ( \lambda \,{\it \_a} \right ) }}{d{\it \_a}}+{\it \_F1} \left ({\frac{x\lambda \,b-a\Ci \left ( \arcsin \left ( \lambda \,y \right ) \right ) }{\lambda \,b}} \right ) \]