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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

Added January 14, 2019.

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

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