Solve $y' + t y = t^{3} y^{3}$ $y'+ty=t^3y^3$ ?

Question

Solve $y' + t y = t^{3} y^{3}$ $y'+ty=t^3y^3$ ?

1 Answer

Impact of this question

1825 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-13T17:06:05

See below.

Explanation:

Making the substitution

$y = \frac{t}{z} \Rightarrow \frac{1}{2} t \frac{d}{d t} (z^{2}) - (t^{2} + 1) z^{2} + t^{6} = 0$ $y = t/z rArr 1/2t d/(dt)(z^2)-(t^2+1)z^2+t^6=0$ now considering

$ξ = z^{2}$ $xi = z^2$ we have the linear differential equation

$\frac{1}{2} t ξ' - (t^{2} + 1) ξ + t^{6} = 0$ $1/2 t xi'-(t^2+1) xi+t^6 = 0$

now solving for $ξ$ $xi$

$ξ = t^{2} (C_{0} e^{t^{2}} + t^{2} + 1)$ $xi = t^2(C_0 e^(t^2)+t^2+1)$ and then

$z = | t | \sqrt{C_{0} e^{t^{2}} + t^{2} + 1}$ $z = abst sqrt(C_0 e^(t^2)+t^2+1)$ and finally

$y = \frac{t}{| t | \sqrt{C_{0} e^{t^{2}} + t^{2} + 1}}$ $y = t/(abs t sqrt(C_0 e^(t^2)+t^2+1))$

Science

Math

Humanities

... and beyond

Solve $y' + t y = t^{3} y^{3}$ $y'+ty=t^3y^3$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

Solve y'+ty=t3y3y'+ty=t^3y^3?

1 Answer

Explanation:

Related questions

Impact of this question

Solve $y' + t y = t^{3} y^{3}$ $y'+ty=t^3y^3$ ?