### 48 HFOPDE, chapter 2.5.2

48.1 problem number 1
48.2 problem number 2
48.3 problem number 3
48.4 problem number 4
48.5 problem number 5
48.6 problem number 6
48.7 problem number 7
48.8 problem number 8
48.9 problem number 9
48.10 problem number 10
48.11 problem number 11
48.12 problem number 12
48.13 problem number 13
48.14 problem number 14
48.15 problem number 15
48.16 problem number 16
48.17 problem number 17
48.18 problem number 18
48.19 problem number 19
48.20 problem number 20
48.21 problem number 21
48.22 problem number 22
48.23 problem number 23

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#### 48.1 problem number 1

problem number 409

Problem 2.5.2.1 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + a x^n \ln ^k(\lambda y) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{\frac{n (-\log (\lambda y))^k \log ^{-k}(\lambda y) \text{Gamma}(1-k,-\log (\lambda y))}{\lambda }+\frac{(-\log (\lambda y))^k \log ^{-k}(\lambda y) \text{Gamma}(1-k,-\log (\lambda y))}{\lambda }-a x^{n+1}}{n+1}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{x}^{n+1}a-n\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y}{a}} \right )$

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#### 48.2 problem number 2

problem number 410

Problem 2.5.2.2 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + a y^n \ln ^k(\lambda x) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (-\frac{y^{-n} (-\log (\lambda x))^{-k} \left (-a y^n \log ^k(\lambda x) \text{Gamma}(k+1,-\log (\lambda x))+a n y^n \log ^k(\lambda x) \text{Gamma}(k+1,-\log (\lambda x))+\lambda y (-\log (\lambda x))^k\right )}{\lambda (n-1)}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{y}{{y}^{n}}}+an\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x-a\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x \right )$

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#### 48.3 problem number 3

problem number 411

Problem 2.5.2.3 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left (y^2+ a \ln (\beta x) y - a b \ln (\beta x) - b^2 \right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}+y\int \! \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}\,{\rm d}x-b\int \! \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}\,{\rm d}x}{-b+y}} \right )$

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#### 48.4 problem number 4

problem number 412

Problem 2.5.2.4 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left (y^2+ a x \ln ^m(b x) y + a \ln ^m(b x) \right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac{a \left ( \ln \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{a \left ( \ln \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac{a \left ( \ln \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right )$

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#### 48.5 problem number 5

problem number 413

Problem 2.5.2.5 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left (a x^n y^2- a b x^{n+1} y \ln (x) + b \ln (x) + b \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 48.6 problem number 6

problem number 414

Problem 2.5.2.6 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x - \left ((n+1)x^n y^2 - a x^{n+1}(\ln x)^m y + a(\ln x)^m \right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{y{x}^{n+1}-1} \left ( y{x}^{n+1}\int \!{\frac{{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}xn+y{x}^{n+1}\int \!{\frac{{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}x-{{\rm e}^{\int \!{\frac{a{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}x-2\,n-2}{x}}\,{\rm d}x}}{x}^{n+1}-\int \!{\frac{{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}xn-\int \!{\frac{{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}x \right ) } \right )$

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#### 48.7 problem number 7

problem number 415

Problem 2.5.2.7 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left (a (\ln x)^n y^2 + b m x^{m-1} - a b^2 x^{2 m} (\ln x)^n \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 48.8 problem number 8

problem number 416

Problem 2.5.2.8 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left (a (\ln x)^n y^2 - a b x y(\ln x)^{n+1} + b \ln x+ b \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 48.9 problem number 9

problem number 417

Problem 2.5.2.9 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left (a (\ln x)^k (y - b x^n-c)^3 + b n x^{n-1} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{(-\log (x))^{-k} \left (2 a b^2 x^{2 n} \log ^k(x) \text{Gamma}(k+1,-\log (x))+4 a b c x^n \log ^k(x) \text{Gamma}(k+1,-\log (x))-4 a b y x^n \log ^k(x) \text{Gamma}(k+1,-\log (x))+2 a c^2 \log ^k(x) \text{Gamma}(k+1,-\log (x))-4 a c y \log ^k(x) \text{Gamma}(k+1,-\log (x))+2 a y^2 \log ^k(x) \text{Gamma}(k+1,-\log (x))+(-\log (x))^k\right )}{\left (b x^n+c-y\right )^2}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{2\,{x}^{2\,n}a{b}^{2}\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x-4\,y{x}^{n}ab\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+4\,{x}^{n}abc\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+2\,{y}^{2}a\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x-4\,yac\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+2\,a{c}^{2}\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+1}{ \left ( y-b{x}^{n}-c \right ) ^{2}}} \right )$

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#### 48.10 problem number 10

problem number 418

Problem 2.5.2.10 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left (a (\ln x)^n y^2 + b(\ln x)^m y+ b c (\ln x)^m - a c^2 (\ln x)^n \right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{ya\int \! \left ( \ln \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,ca\int \! \left ( \ln \left ( x \right ) \right ) ^{n}\,{\rm d}x+b\int \! \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x}}\,{\rm d}x+a\int \! \left ( \ln \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,ca\int \! \left ( \ln \left ( x \right ) \right ) ^{n}\,{\rm d}x+b\int \! \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x}}\,{\rm d}xc+{{\rm e}^{-2\,ca\int \! \left ( \ln \left ( x \right ) \right ) ^{n}\,{\rm d}x+b\int \! \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x}}}{c+y}} \right )$

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#### 48.11 problem number 11

problem number 419

Problem 2.5.2.11 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + \left (a y+ b \ln x \right )^2 w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{a^3 y \sqrt{\frac{b}{a^3}}+a^2 b \sqrt{\frac{b}{a^3}} \log (x)}{b}\right )-a^2 \sqrt{\frac{b}{a^3}} \log (x)\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{a\sqrt{ab}} \left ( -\ln \left ( x \right ) \sqrt{ab}+\arctan \left ({\frac{a \left ( ay+b\ln \left ( x \right ) \right ) }{\sqrt{ab}}} \right ) \right ) } \right )$

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#### 48.12 problem number 12

problem number 420

Problem 2.5.2.12 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + \left (x y^2 - A^2 x (\ln \beta x)^2 + A \right ) w_y = 0$

Mathematica

$\text{DSolve}\left [w^{(0,1)}(x,y) \left (-A^2 x \log ^2(\beta x)+A+x y^2\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ]$

Maple

$\text{ sol=() }$

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#### 48.13 problem number 13

problem number 421

Problem 2.5.2.13 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + \left (x y^2 - A^2 x (\ln (\beta x))^{2 k} + k A (\ln (\beta x))^{k-1} \right ) w_y = 0$

Mathematica

$\text{DSolve}\left [w^{(0,1)}(x,y) \left (-A^2 x \log ^{2 k}(\beta x)+A k \log ^{k-1}(\beta x)+x y^2\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ]$

Maple

$\text{ sol=() }$

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#### 48.14 problem number 14

problem number 422

Problem 2.5.2.14 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + \left (a x^n y^2 + b - a b^2 x^n (\ln x)^2 \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 48.15 problem number 15

problem number 423

Problem 2.5.2.15 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + \left ( a (\ln (\lambda x))^m y^2 + k y+ a b^2 x^{2 k} (\ln (\lambda x))^m \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{y x^{-k}}{\sqrt{b^2}}\right )-\frac{a \sqrt{b^2} x^k (\lambda x)^{-k} \log ^m(\lambda x) (-k \log (\lambda x))^{-m} \text{Gamma}(m+1,-k \log (\lambda x))}{k}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( ab\int \!{x}^{k-1} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{m}\,{\rm d}x-\arctan \left ({\frac{{x}^{-k}y}{b}} \right ) \right )$

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#### 48.16 problem number 16

problem number 424

Problem 2.5.2.16 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + \left ( a x^n(y + b \ln x)^2 - b \right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ab\ln \left ( x \right ){x}^{n}+a{x}^{n}y+n}{n \left ( y+b\ln \left ( x \right ) \right ) }} \right )$

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#### 48.17 problem number 17

problem number 425

Problem 2.5.2.17 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + \left ( a x^{2 n} \ln (x) y^2 + (b x^n \ln x - n) y + c \ln x \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{\left (\sqrt{b^2-4 a c}+2 a y x^n+b\right ) \exp \left (\frac{\sqrt{a} \sqrt{c} x^n \left (\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}+\frac{b}{\sqrt{a} \sqrt{c}}\right ) (n \log (x)-1)}{2 n^2}-\frac{\sqrt{a} \sqrt{c} x^n \left (\frac{b}{\sqrt{a} \sqrt{c}}-\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}\right ) (n \log (x)-1)}{2 n^2}\right )}{\sqrt{b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{n}^{2}} \left ( -\ln \left ( x \right ){x}^{n}\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }n+2\,b{n}^{2}\arctan \left ({\frac{b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt{4\,ac{b}^{2}-{b}^{4}}}} \right ) +\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n} \right ) } \right )$

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#### 48.18 problem number 18

problem number 426

Problem 2.5.2.18 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x^k w_x + \left (a y^n (\ln x)^m + b y (\ln x)^s \right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

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

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#### 48.19 problem number 19

problem number 427

Problem 2.5.2.19 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(a \ln x+b) w_x + \left (y^2+ c(\ln x)^n y- \lambda ^2 + \lambda c( \ln x)^n \right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( \int \!{\frac{1}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!-{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}+y\int \!{\frac{1}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}} \right ) \left ({{\rm e}^{\int \!-{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}\lambda +y \right ) ^{-1}} \right )$

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#### 48.20 problem number 20

problem number 428

Problem 2.5.2.20 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(a \ln x+b) w_x + \left ((\ln x)^n y^2- c y - \lambda ^2 ( \ln x)^n + c \lambda \right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( -\int \!{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac{2\,\lambda \, \left ( \ln \left ( x \right ) \right ) ^{n}-c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!{\frac{2\,\lambda \, \left ( \ln \left ( x \right ) \right ) ^{n}-c}{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!-{\frac{2\,\lambda \, \left ( \ln \left ( x \right ) \right ) ^{n}-c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}+y\int \!{\frac{ \left ( \ln \left ( x \right ) \right ) ^{n}}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac{2\,\lambda \, \left ( \ln \left ( x \right ) \right ) ^{n}-c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{2\,\lambda \, \left ( \ln \left ( x \right ) \right ) ^{n}-c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}} \right ) \left ( -{{\rm e}^{\int \!{\frac{2\,\lambda \, \left ( \ln \left ( x \right ) \right ) ^{n}-c}{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!-{\frac{2\,\lambda \, \left ( \ln \left ( x \right ) \right ) ^{n}-c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}\lambda +y \right ) ^{-1}} \right )$

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#### 48.21 problem number 21

problem number 429

Problem 2.5.2.21 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x^2 \ln (a x) w_x - \left ( x^2 y^2 \ln (a x) + 1\right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{xy\ln \left ( ax \right ) -1}{\ln \left ( ax \right ) y\Ei \left ( 1,-\ln \left ( ax \right ) \right ) x+a{x}^{2}y-\Ei \left ( 1,-\ln \left ( ax \right ) \right ) }} \right )$

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#### 48.22 problem number 22

problem number 430

Problem 2.5.2.22 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$\ln ^k(\lambda x) w_x + \left ( a y^n + b y \ln ^m x\right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

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

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#### 48.23 problem number 23

problem number 431

Problem 2.5.2.23 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$\ln ^k(\lambda x) w_x + \left ( a y^n \ln ^m x + b y \right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

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