### 99 HFOPDE, chapter 4.2.4

99.1 Problem 1

99.5 Problem 3
99.6 Problem 4
99.7 Problem 5
99.8 Problem 6
99.9 Problem 7
99.10 Problem 8
99.11 Problem 9
99.12 Problem 10
99.13 Problem 11
99.14 Problem 12
99.15 Problem 13

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#### 99.1 Problem 1

problem number 837

Problem Chapter 4.2.4.1 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = (c x^n + d y^m) w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{a y-b x}{a}\right ) \exp \left (\frac{d \left (\frac{a y-b x}{b (m+1)}+\frac{b x}{b m+b}\right ) \left (\frac{a y-b x}{a}+\frac{b x}{a}\right )^m}{a}+\frac{c x^{n+1}}{a (n+1)}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){{\rm e}^{{\frac{1}{a \left ( n+1 \right ) b \left ( m+1 \right ) } \left ( c{x}^{n+1}bm+d \left ({\frac{bx}{a}}+{\frac{ay-bx}{a}} \right ) ^{m+1}an+{x}^{n+1}cb+d \left ({\frac{bx}{a}}+{\frac{ay-bx}{a}} \right ) ^{m+1}a \right ) }}}$

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#### 99.2 Problem 2 case $$n\neq -1, n\neq -2$$

problem number 838

Problem Chapter 4.2.4.2 case $$n eq -1, n eq -2$$, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = c x^n y w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \exp \left (\frac{c x^{n+1} (a (n+2) y-b x)}{a^2 (n+1) (n+2)}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){{\rm e}^{{\frac{c \left ( ay \left ( 2+n \right ){x}^{n+1}-{x}^{2+n}b \right ) }{ \left ( 2+n \right ) \left ( n+1 \right ){a}^{2}}}}}$

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#### 99.3 Problem 2 case $$n = -1$$

problem number 839

Problem Chapter 4.2.4.2 case $$n= -1$$, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = c x^n y w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{a y-b x}{a}\right ) e^{\frac{c (\log (x) (a y-b x)+b x)}{a^2}}\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){x}^{{\frac{c \left ( ay-bx \right ) }{{a}^{2}}}}{{\rm e}^{{\frac{bcx}{{a}^{2}}}}}$

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#### 99.4 Problem 2 case $$n = -2$$

problem number 840

Problem Chapter 4.2.4.2 case $$n= -2$$, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = c x^n y w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{a y-b x}{a}\right ) e^{\frac{c \left (b \log (x)-\frac{a y-b x}{x}\right )}{a^2}}\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){x}^{{\frac{cb}{{a}^{2}}}}{{\rm e}^{-{\frac{c \left ( ay-bx \right ) }{{a}^{2}x}}}}$

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#### 99.5 Problem 3

problem number 841

Problem Chapter 4.2.4.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + y w_y = a(x^2+y^2)^k w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{y}{x}\right ) e^{\frac{a \left (x^2 \left (\frac{y^2}{x^2}+1\right )\right )^k}{2 k}}\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{y}{x}} \right ){{\rm e}^{1/2\,{\frac{a}{k} \left ({x}^{2} \left ({\frac{{y}^{2}}{{x}^{2}}}+1 \right ) \right ) ^{k}}}}$

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#### 99.6 Problem 4

problem number 842

Problem Chapter 4.2.4.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a x w_x + b y w_y = c x^n y^m w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac{b}{a}}\right ) e^{\frac{c y^m x^n}{a \left (\frac{b m}{a}+n\right )}}\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ){{\rm e}^{{\frac{c{x}^{n}{y}^{m}}{an+mb}}}}$

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#### 99.7 Problem 5

problem number 843

Problem Chapter 4.2.4.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a x w_x + b y w_y = (c x^n + k y^m) w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac{b}{a}}\right ) e^{\frac{c x^n}{a n}+\frac{k y^m}{b m}}\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ){{\rm e}^{{\frac{c{x}^{n}mb+k{y}^{m}an}{anmb}}}}$

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#### 99.8 Problem 6

problem number 844

Problem Chapter 4.2.4.6, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$m x w_x + n y w_y = (a x^n + b y^m)^k w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac{n}{m}}\right ) e^{\frac{\left (a x^n+b y^m\right )^k}{k m n}}\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac{n}{m}}} \right ){{\rm e}^{{\frac{ \left ( a{x}^{n}+b{y}^{m} \right ) ^{k}}{knm}}}}$

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#### 99.9 Problem 7

problem number 845

Problem Chapter 4.2.4.7, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a x^n w_x + b y^m w_y = (c x^k + d y^s) w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{b x^{1-n}}{a (n-1)}-\frac{y^{1-m}}{m-1}\right ) \exp \left (\frac{c x^{k-n+1}}{a k-a n+a}-\frac{d y^{1-m} \left (\left (y^{m-1}\right )^{\frac{1}{m-1}}\right )^s}{b (m-s-1)}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-{x}^{1-n}b \left ( m-1 \right ) + \left ( n-1 \right ) a{y}^{-m+1}}{ \left ( n-1 \right ) a}} \right ){{\rm e}^{{\frac{1}{b \left ( -s+m-1 \right ) a \left ( k-n+1 \right ) } \left ( -{y}^{-m+1} \left ( n-1 \right ) ^{{\frac{s}{m-1}}}{a}^{{\frac{s+m-1}{m-1}}} \left ( \left ( n-1 \right ) a{y}^{-m+1} \right ) ^{-{\frac{s}{m-1}}}d \left ( k-n+1 \right ){{\rm e}^{{\frac{i/2\pi \,s}{m-1} \left ( -{\it csgn} \left ( i \left ( n-1 \right ){y}^{-m+1}a \right ) -{\it csgn} \left ({\frac{i}{n-1}} \right ) +{\it csgn} \left ( i{y}^{-m+1}a \right ){\it csgn} \left ( i \left ( n-1 \right ){y}^{-m+1}a \right ){\it csgn} \left ({\frac{i}{n-1}} \right ) +{\it csgn} \left ( i{y}^{-m+1}a \right ){\it csgn} \left ( i{y}^{-m+1} \right ){\it csgn} \left ({\frac{i}{a}} \right ) +{\it csgn} \left ( i{y}^{-m+1} \right ) -{\it csgn} \left ({\frac{i}{a}} \right ) \right ) }}}+{x}^{k-n+1}cb \left ( -s+m-1 \right ) \right ) }}}$

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#### 99.10 Problem 8

problem number 846

Problem Chapter 4.2.4.8, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a x^n w_x + b x^m y w_y = (c x^k y^s + d) w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (y e^{-\frac{b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\frac{x^{1-n} \left (\frac{d}{1-n}-\frac{c x^k y^s e^{-\frac{b s x^{m-n+1}}{a m-a n+a}} \left (-\frac{b s x^{m-n+1}}{a m-a n+a}\right )^{\frac{-k+n-1}{m-n+1}} \text{Gamma}\left (\frac{k-n+1}{m-n+1},-\frac{b s x^{m-n+1}}{a m-a n+a}\right )}{m-n+1}\right )}{a}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ({{\it \_a}}^{k-n}c \left ( y{{\rm e}^{-{\frac{b \left ({x}^{m-n+1}-{{\it \_a}}^{m-n+1} \right ) }{a \left ( m-n+1 \right ) }}}} \right ) ^{s}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}}}$

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#### 99.11 Problem 9

problem number 847

Problem Chapter 4.2.4.9, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a x^n w_x + (b x^m y+c x^k) w_y = (s x^p y^q + d) w$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( -a{{\rm e}^{-1/2\,{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \left ({\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{2\,m-2\,n+2}}}c{x}^{k-m} \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ({\frac{k+m-2\,n+2}{2\,m-2\,n+2}},{\frac{k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) - \left ({\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) ^{2}{{\rm e}^{-1/2\,{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \left ({x}^{k-n+1}b+{x}^{k-m}a \left ( k+m-2\,n+2 \right ) \right ) \WhittakerM \left ({\frac{k-m}{2\,m-2\,n+2}},{\frac{k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) +{{\rm e}^{-{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}}}aby \left ( k-n+1 \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ) } \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( s{{\it \_a}}^{-n+p} \left ({\frac{1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( a{{\it \_a}}^{k-m}{{\rm e}^{1/2\,{\frac{b{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \left ({\frac{b{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ({\frac{k+m-2\,n+2}{2\,m-2\,n+2}},{\frac{k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac{b{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) -a{x}^{k-m}{{\rm e}^{-1/2\,{\frac{b \left ( -2\,{{\it \_a}}^{m-n+1}+{x}^{m-n+1} \right ) }{a \left ( m-n+1 \right ) }}}} \left ({\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ({\frac{k+m-2\,n+2}{2\,m-2\,n+2}},{\frac{k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) + \left ( b{{\it \_a}}^{k-n+1}+{{\it \_a}}^{k-m}a \left ( k+m-2\,n+2 \right ) \right ){{\rm e}^{1/2\,{\frac{b{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }}}}c \left ({\frac{b{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{2\,m-2\,n+2}}} \left ( m-n+1 \right ) ^{2} \WhittakerM \left ({\frac{k-m}{2\,m-2\,n+2}},{\frac{k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac{b{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) -{{\rm e}^{-1/2\,{\frac{b \left ( -2\,{{\it \_a}}^{m-n+1}+{x}^{m-n+1} \right ) }{a \left ( m-n+1 \right ) }}}} \left ({\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) ^{2} \left ({x}^{k-n+1}b+{x}^{k-m}a \left ( k+m-2\,n+2 \right ) \right ) \WhittakerM \left ({\frac{k-m}{2\,m-2\,n+2}},{\frac{k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac{b{x}^{m-n+1}}{a \left ( m-n+1 \right ) }} \right ) +{{\rm e}^{-{\frac{b \left ({x}^{m-n+1}-{{\it \_a}}^{m-n+1} \right ) }{a \left ( m-n+1 \right ) }}}}yab \left ( k-n+1 \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ) } \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}}}$

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#### 99.12 Problem 10

problem number 848

Problem Chapter 4.2.4.10, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a x^n w_x + b x^m y^k w_y = (c x^p y^q + s) w$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b \left ( k-1 \right ){x}^{m-n+1}+{y}^{-k+1}a \left ( m-n+1 \right ) }{a \left ( m-n+1 \right ) }} \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ({{\it \_a}}^{-n+p}c \left ( \left ({\frac{-b \left ( k-1 \right ){{\it \_a}}^{m-n+1}+b \left ( k-1 \right ){x}^{m-n+1}+{y}^{-k+1}a \left ( m-n+1 \right ) }{a \left ( m-n+1 \right ) }} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) ^{q}+{{\it \_a}}^{-n}s \right ) }{d{\it \_a}}}}$

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#### 99.13 Problem 11

problem number 849

Problem Chapter 4.2.4.11, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a y^k w_x + b x^n w_y = (c x^m + s) w$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{y^{k+1}}{k+1}-\frac{b x^{n+1}}{a n+a}\right ) \exp \left (\frac{x \left (\left (y^{-k-1}\right )^{-\frac{1}{k+1}}\right )^{-k} \left (\frac{a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac{k}{k+1}} \left (c x^m \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{m+1}{n+1},\frac{m+n+2}{n+1},\frac{b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )+(m+1) s \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{1}{n+1},\frac{1}{n+1}+1,\frac{b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )\right )}{a (m+1)}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ){{\rm e}^{\int ^{x}\!{\frac{c{{\it \_a}}^{m}+s}{a} \left ( \left ({\frac{b \left ( k+1 \right ){{\it \_a}}^{n+1}-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}}}$

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#### 99.14 Problem 12

problem number 850

Problem Chapter 4.2.4.12, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x(x^n + (a n -1) y^n) w_x + y(y^n + (a n -1) x^n) w_y = k n (x^n + y^n) w$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( \left ( -{y}^{n}{x}^{-{a}^{-1}}+{x}^{{\frac{an-1}{a}}} \right ) \left ({y}^{n} \right ) ^{-{\frac{1}{an}}} \right ){{\rm e}^{\int ^{x}\!{\frac{kn}{{\it \_a}} \left ({{\it \_a}}^{n}+ \left ( \RootOf \left ( -{y}^{n}{x}^{-{a}^{-1}} \left ({y}^{n} \right ) ^{-{\frac{1}{an}}}\sqrt [a]{{\it \_a}} \left ({{\it \_Z}}^{n} \right ) ^{{\frac{1}{an}}}+{x}^{{\frac{an-1}{a}}} \left ({y}^{n} \right ) ^{-{\frac{1}{an}}}\sqrt [a]{{\it \_a}} \left ({{\it \_Z}}^{n} \right ) ^{{\frac{1}{an}}}+{{\it \_Z}}^{n}-{{\it \_a}}^{n} \right ) \right ) ^{n} \right ) \left ( \left ( \RootOf \left ( -{y}^{n}{x}^{-{a}^{-1}} \left ({y}^{n} \right ) ^{-{\frac{1}{an}}}\sqrt [a]{{\it \_a}} \left ({{\it \_Z}}^{n} \right ) ^{{\frac{1}{an}}}+{x}^{{\frac{an-1}{a}}} \left ({y}^{n} \right ) ^{-{\frac{1}{an}}}\sqrt [a]{{\it \_a}} \left ({{\it \_Z}}^{n} \right ) ^{{\frac{1}{an}}}+{{\it \_Z}}^{n}-{{\it \_a}}^{n} \right ) \right ) ^{n}an+{{\it \_a}}^{n}- \left ( \RootOf \left ( -{y}^{n}{x}^{-{a}^{-1}} \left ({y}^{n} \right ) ^{-{\frac{1}{an}}}\sqrt [a]{{\it \_a}} \left ({{\it \_Z}}^{n} \right ) ^{{\frac{1}{an}}}+{x}^{{\frac{an-1}{a}}} \left ({y}^{n} \right ) ^{-{\frac{1}{an}}}\sqrt [a]{{\it \_a}} \left ({{\it \_Z}}^{n} \right ) ^{{\frac{1}{an}}}+{{\it \_Z}}^{n}-{{\it \_a}}^{n} \right ) \right ) ^{n} \right ) ^{-1}}{d{\it \_a}}}}$

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#### 99.15 Problem 13

problem number 851

Problem Chapter 4.2.4.13, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x((n-2) y^n -2 x^n ) w_x + y(2 y^n - (n-2) x^n) w_y =\left ( (a(n-2)+2 b) y^n - (2 a + b(n-2)) x^n) \right ) w$

Mathematica

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

Maple

$\text{ sol=() }$ Contains RootOf