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Sergio [31]
3 years ago
8

Use an appropriate substitution to solve the equation

Mathematics
1 answer:
Yuri [45]3 years ago
4 0
y'-\dfrac8xy=\dfrac{y^5}{x^{20}}

This ODE is in standard Bernoulli form, so we can divide through both sides by y^5 to get

y^{-5}y'-\dfrac8xy^{-4}=\dfrac1{x^{20}}

Substitute z=y^{-4}, so that z'=-4y^{-5}y'. Then the ODE can be written as a linear one in z:

-\dfrac14z'-\dfrac8xz=\dfrac1{x^{20}}
z'+\dfrac{32}xz=-\dfrac4{x^{20}}

Multiplying both sides by x^{32} gives

x^{32}z'+32x^{31}z=-4x^{12}
(x^{32}z)'=-4x^{12}
x^{32}z=\displaystyle-4\int x^{12}\,\mathrm dx
x^{32}z=-\dfrac4{13}x^{13}+C
z=-\dfrac4{13x^{19}}+\dfrac C{x^{32}}

Back-substitute to solve for y:

\dfrac1{y^4}=-\dfrac4{13x^{19}}+\dfrac C{x^{32}}
y^4=\dfrac1{Cx^{-32}-\frac4{13}x^{-19}}
y=\left(\dfrac{x^{32}}{C-\frac4{13}x^{13}}\right)^{1/4}
y=\dfrac{x^8}{(C-\frac4{13}x^{13})^{1/4}}

Given that y(1)=1, you have

1=\dfrac{1^8}{(C-\frac4{13}1^{13})^{1/4}}
1=\dfrac1{(C-\frac4{13})^{1/4}}
1=\left(C-\dfrac4{13}\right)^{1/4}
1=C-\dfrac4{13}
C=\dfrac{17}{13}

so that the particular solution is

y=\dfrac{x^8}{(\frac{17}{13}-\frac4{13}x^{13})^{1/4}}
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