58 HFOPDE, chapter 2.8.1

 58.1 problem number 1
 58.2 problem number 2
 58.3 problem number 3
 58.4 problem number 4
 58.5 problem number 5
 58.6 problem number 6
 58.7 problem number 7
 58.8 problem number 8
 58.9 problem number 9
 58.10 problem number 10
 58.11 problem number 11
 58.12 problem number 12
 58.13 problem number 13

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58.1 problem number 1

problem number 544

Added Feb. 4, 2019.

Problem 2.8.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y+g(x) \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-e^{-\int _1^x f(K[1]) \, dK[1]} \left (e^{\int _1^x f(K[1]) \, dK[1]} \int _1^x g(K[2]) e^{-\text{Integrate}[f(K[1]),\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}]} \, dK[2]-y\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!g \left ( x \right ){{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}} \right ) \]

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58.2 problem number 2

problem number 545

Added Feb. 4, 2019.

Problem 2.8.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y+g(x) y^k \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y^{-k} e^{-\int _1^x f(K[1]) \, dK[1]} \left (y^k \left (-e^{\int _1^x f(K[1]) \, dK[1]}\right ) \left (\int _1^x g(K[2]) \exp (-(1-k) \text{Integrate}[f(K[1]),\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}]) \, dK[2]\right )+k y^k e^{\int _1^x f(K[1]) \, dK[1]} \left (\int _1^x g(K[2]) \exp (-(1-k) \text{Integrate}[f(K[1]),\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}]) \, dK[2]\right )+y e^{k \int _1^x f(K[1]) \, dK[1]}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f \left ( x \right ) \,{\rm d}x}}+k\int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f \left ( x \right ) \,{\rm d}x}}g \left ( x \right ) \,{\rm d}x-\int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f \left ( x \right ) \,{\rm d}x}}g \left ( x \right ) \,{\rm d}x \right ) \]

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58.3 problem number 3

problem number 546

Added Feb. 4, 2019.

Problem 2.8.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+f(x) y -a^2 -a f(x)\right ) w_y = 0 \]

Mathematica

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

Maple

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

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58.4 problem number 4

problem number 547

Added Feb. 4, 2019.

Problem 2.8.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+x f(x) y + f(x)\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac{f \left ( x \right ){x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{f \left ( x \right ){x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac{f \left ( x \right ){x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]

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58.5 problem number 5

problem number 548

Added Feb. 4, 2019.

Problem 2.8.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x - \left ( (k+1)x^k y^2-x^{k+1} f(x) y+f(x)\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{{x}^{k+1}y-1} \left ( -y{x}^{k+1}\int \!{\frac{{{\rm e}^{\int \!{x}^{k+1}f \left ( x \right ) \,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk-y{x}^{k+1}\int \!{\frac{{{\rm e}^{\int \!{x}^{k+1}f \left ( x \right ) \,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{{x}^{k+1}f \left ( x \right ) x-2\,k-2}{x}}\,{\rm d}x}}{x}^{k+1}+\int \!{\frac{{{\rm e}^{\int \!{x}^{k+1}f \left ( x \right ) \,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk+\int \!{\frac{{{\rm e}^{\int \!{x}^{k+1}f \left ( x \right ) \,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \right ) } \right ) \]

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58.6 problem number 6

problem number 549

Added Feb. 4, 2019.

Problem 2.8.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+a y-a b- b^2 f(x)\right ) w_y = 0 \]

Mathematica

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

Maple

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

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58.7 problem number 7

problem number 550

Added Feb. 4, 2019.

Problem 2.8.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f[x] y^2-a x^n f[x] y+a n x^{n-1}\right ) w_y = 0 \]

Mathematica

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

Maple

\[ \text{ sol=() } \]

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58.8 problem number 8

problem number 551

Added Feb. 4, 2019.

Problem 2.8.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+a n x^{n-1}-a^2 x^{2 n} f(x)\right ) w_y = 0 \]

Mathematica

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

Maple

\[ \text{ sol=() } \]

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58.9 problem number 9

problem number 552

Added Feb. 4, 2019.

Problem 2.8.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+g(x) y-a^2 f(x)-a g(x)\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{{\rm e}^{\int \!g \left ( x \right ) \,{\rm d}x+2\,a\int \!f \left ( x \right ) \,{\rm d}x}}+y\int \!{{\rm e}^{\int \!g \left ( x \right ) \,{\rm d}x+2\,a\int \!f \left ( x \right ) \,{\rm d}x}}f \left ( x \right ) \,{\rm d}x-a\int \!{{\rm e}^{\int \!g \left ( x \right ) \,{\rm d}x+2\,a\int \!f \left ( x \right ) \,{\rm d}x}}f \left ( x \right ) \,{\rm d}x}{-a+y}} \right ) \]

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58.10 problem number 10

problem number 553

Added Feb. 4, 2019.

Problem 2.8.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+g(x) y+a n x^{n-1} - a x^n g(x)-a^2 x^{2 n} f(x)\right ) w_y = 0 \]

Mathematica

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

Maple

\[ \text{ sol=() } \]

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58.11 problem number 11

problem number 554

Added Feb. 4, 2019.

Problem 2.8.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2-a x^n g(x) y+a n x^{n-1}+a^2 x^{2 n}(g(x)-f(x))\right ) w_y = 0 \]

Mathematica

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

Maple

\[ \text{ sol=() } \]

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58.12 problem number 12

problem number 555

Added Feb. 4, 2019.

Problem 2.8.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( f(x) y^2+n y+a x^{2 n} f(x)\right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{y x^{-n}}{\sqrt{a}}\right )-\sqrt{a} \int _1^x f(K[1]) K[1]^{n-1} \, dK[1]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \sqrt{a}\int \!{x}^{n-1}f \left ( x \right ) \,{\rm d}x-\arctan \left ({\frac{{x}^{-n}y}{\sqrt{a}}} \right ) \right ) \]

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58.13 problem number 13

problem number 556

Added Feb. 4, 2019.

Problem 2.8.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( x^{2 n} f(x) y^2+(a x^n f(x)-n) y+b f(x)\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{a}{\sqrt{{a}^{2} \left ({a}^{2}-4\,b \right ) }} \left ( -\sqrt{{a}^{2} \left ({a}^{2}-4\,b \right ) }\int \!{\frac{{x}^{n}f \left ( x \right ) }{x}}\,{\rm d}x-2\,a\arctanh \left ({\frac{ \left ( 2\,{x}^{n}y+a \right ) a}{\sqrt{{a}^{2} \left ({a}^{2}-4\,b \right ) }}} \right ) \right ) } \right ) \]