What is a solution to the differential equation $\frac{dy}{dx}=12{x}^{3}y$ with the particular solution y(0)=2?

Darius Nash
2022-09-13
Answered

What is a solution to the differential equation $\frac{dy}{dx}=12{x}^{3}y$ with the particular solution y(0)=2?

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I'm currently stuck with a problem where I'm supposed to find all solutions that are asymptotic to the line $y=3-t$ when $t\to \mathrm{\infty}$. This is the demand, from here I'm supposed to create a first order linear differential equation. Can someone help me get started with this problem? Unsure of how to start....

Asymptotic would mean that for example $x(t)=y(t)-1/t$ would satisfy the given demand, not sure how to go further with this although.

A first order linear differential equation means I should have some sort of connection between my function and the derivative of the function, I cant make that connection....

It's my first time using this forum so I've probably made every mistake you can make, hopefully my question is still relevant...

Asymptotic would mean that for example $x(t)=y(t)-1/t$ would satisfy the given demand, not sure how to go further with this although.

A first order linear differential equation means I should have some sort of connection between my function and the derivative of the function, I cant make that connection....

It's my first time using this forum so I've probably made every mistake you can make, hopefully my question is still relevant...

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What is a solution to the differential equation $x\frac{dy}{dx}=\frac{1}{y}$?

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I need to find a series solution to the following simple differential equation

${x}^{2}{y}^{\prime}=y$

Assuming the solution to be of the form $y=\sum {a}_{n}{x}^{n}$ and equating the coefficients on both the sides, all the coefficients turn out to be zero which is definitely wrong.

Any help is appreciated.

${x}^{2}{y}^{\prime}=y$

Assuming the solution to be of the form $y=\sum {a}_{n}{x}^{n}$ and equating the coefficients on both the sides, all the coefficients turn out to be zero which is definitely wrong.

Any help is appreciated.

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