### 62 HFOPDE, chapter 2.8.5

62.1 problem number 1
62.2 problem number 2
62.3 problem number 3
62.4 problem number 4
62.5 problem number 5

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#### 62.1 problem number 1

problem number 575

Problem 2.8.5.1 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( \lambda \sin (\lambda x) y^2 + f(x) \cos (\lambda x) y-f(x) \right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({(\cos \left ( \lambda \,x \right ) y-1) \left ( -\cos \left ( \lambda \,x \right ) \int \!-\lambda \,{{\rm e}^{\int \!{\frac{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\cos \left ( \lambda \,x \right ) \sin \left ( \lambda \,x \right ) }}\,{\rm d}x}}\sin \left ( \lambda \,x \right ) \,{\rm d}xy+{{\rm e}^{\int \!{\frac{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\cos \left ( \lambda \,x \right ) \sin \left ( \lambda \,x \right ) }}\,{\rm d}x}}\cos \left ( \lambda \,x \right ) +\int \!-\lambda \,{{\rm e}^{\int \!{\frac{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\cos \left ( \lambda \,x \right ) \sin \left ( \lambda \,x \right ) }}\,{\rm d}x}}\sin \left ( \lambda \,x \right ) \,{\rm d}x \right ) ^{-1}} \right )$

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#### 62.2 problem number 2

problem number 576

Problem 2.8.5.2 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( f(x) y^2-a^2 f(x)+a \lambda \sin (\lambda x)+a^2 f(x) \sin ^2(\lambda x) \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 62.3 problem number 3

problem number 577

Problem 2.8.5.3 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( f(x) y^2-a^2 f(x)+a \lambda \cos (\lambda x)+a^2 f(x) \cos ^2(\lambda x) \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 62.4 problem number 4

problem number 578

Problem 2.8.5.4 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( f(x) y^2-a(a f(x)-\lambda ) \tan ^2(\lambda x)+a \lambda \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 62.5 problem number 5

problem number 579

Problem 2.8.5.5 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( f(x) y^2-a(a f(x)-\lambda ) \cot ^2(\lambda x)+a \lambda \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$