Find the power series solution of the differential equation given below around the point X0=0. $({x}^{3}-1)({d}^{2}y/d{x}^{2})+{x}^{2}(dy/dx)+xy=0$

ghostmasakii
2022-01-28
Answered

Find the power series solution of the differential equation given below around the point X0=0. $({x}^{3}-1)({d}^{2}y/d{x}^{2})+{x}^{2}(dy/dx)+xy=0$

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Solve the following differential equations: $y-{y}^{\prime}=5{e}^{x}-\mathrm{sin}\left(2x\right)$

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Use the definition of Laplace Transforms to find

Then rewrite f(t) as a sum of step functions,

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To find:

The Laplace transform of$L[{e}^{-3t}{t}^{4}]$

The Laplace transform of

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How to solve $a\ddot{u}+b{\left(\dot{u}\right)}^{2}+\dot{u}+\dot{u}c{e}^{u}+{e}^{u}-{e}^{2u}+1=0$

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Find the laplace transform

$L[({t}^{2}+4){e}^{2t}-{e}^{-2t}\mathrm{cos}t]$

asked 2020-10-31

Find the laplace transform of the following

$f(t)=t{u}_{2}(t)$

Ans.$F(s)=(\frac{1}{{s}^{2}}+\frac{2}{s}){e}^{-2s}$

Ans.

asked 2022-01-16

Suppose we have a mechanical system with 1 degree of freedom, i.e. an ODE

$\left(1\right)\ddot{q}+{V}^{\prime}\left(q\right)=0$ ,

where$V:R\to R$ is some smooth function (potential energy). We then easily see that any solution of this equation must satisfy

$\frac{{\dot{q}}^{2}}{2}+V\left(q\right)=\text{constant}$

In other words, if we put

$E(q,\dot{q})=\frac{{\dot{q}}^{2}}{2}+V\left(q\right)$

(energy), then the image of every solution of (1) must lie in a level set of E.

where

In other words, if we put

(energy), then the image of every solution of (1) must lie in a level set of E.