emancipezN
2021-02-02
Answered

Find the general solution of the second order non-homogeneous linear equation:

${y}^{\u2033}-7{y}^{\prime}+12y=10\mathrm{sin}t+12t+5$

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Willie

Answered 2021-02-03
Author has **95** answers

The given second order differential equation is

Find the general solution of the differential equation as follows.

The corresponding homogeneous equation is

The auxiliary equation of the corresponding homogeneous equation is

Therefore, the roots of the auxiliary equation are

Hence, the complementary solution is

Now obtain the particular solution by method of undetermined coefficients as follows.

The choice for the particular solution is

This particular solution

Therefore,

Equate the coefficients of like terms on both sides and obtain,

Then,

Substitute

Now from third equation

Then,

Hence the particular solution is,

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I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve.

I am aware that the Weierstrass $\mathrm{\wp}$ - Elliptic function satisfies a differential equation. We can then interpret this Differential Equation as an algebraic equation, with solutions found on elliptic curves. However this differential equations is of the first order.

So, are there periodic function(s) $F(x)$ that satisfy a second order differential equation, such that we can say these parametrize an algebraic curve?

Could a Bessel function be one such solution?

I am aware that the Weierstrass $\mathrm{\wp}$ - Elliptic function satisfies a differential equation. We can then interpret this Differential Equation as an algebraic equation, with solutions found on elliptic curves. However this differential equations is of the first order.

So, are there periodic function(s) $F(x)$ that satisfy a second order differential equation, such that we can say these parametrize an algebraic curve?

Could a Bessel function be one such solution?

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I need to solve an equation of this type

${\left(\frac{dy}{dx}\right)}^{2}+\frac{{y}^{2}}{{b}^{2}}={a}^{2}$

but I don't know how to start Any help would be welcome, Thanks !

${\left(\frac{dy}{dx}\right)}^{2}+\frac{{y}^{2}}{{b}^{2}}={a}^{2}$

but I don't know how to start Any help would be welcome, Thanks !

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Solving second-order nonlinear ordinary differential equation

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given that$y\left(0\right)=1\text{}\text{and}\text{}{y}^{\prime}\left(0\right)=\frac{1}{3}$ .

given that

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