## Find the general solution of the simple harmonic oscillator November 27    1

Question: Find the general solution of the simple harmonic oscillator equation, $$\ddot{x}=-\omega^2x$$My answer: $x(t)=A\cos(\omega t)+B\sin(\omega t)$Solution given: $x(t)=x_0\cos(\omega t)+\frac{\dot{x_0}}{\omega}\sin(\omega t)$I can understand in

## solution of a ODE with a funtion of $\dot{x}$ November 27    2

I have the equation: $$m\ddot{x}(t)+kx(t)=A$$ with m, k as constants and $$A = \left\{ \begin{array}{lr} a & : \dot{x}(t) <0\\ -a & : \dot{x}(t) >0 \end{array} \right.$$ a another constant. My question is how to solve for x(t) since A is a

## Solve the differential equation: $\dot{x}(t)^2+x(t)=\sin(x)^2$ November 27    2

I am unable to solve this nonlinear differential equation: $$\dot{x}(t)^2+x(t)=\sin(x)^2$$ I tried with Maple without success. Is it possible to solve it without the use of numerical methods? Thanks.If $x(t)$ is of class $C^2$ you can differentiate t

## Elementary Proof Involving Wronskian and Existence and Uniqueness November 27    1

I'm in need of some help to understand an elementary proof involving the Wronskian of $n$ vector functions and the following existence and uniqueness theorem. Existence and UniquenessSuppose $A(t)$ and $f(t)$ are continuous on an open interval $I$ th

## If Wronskian is zero at some point in an interval, then the Wronskian is zero at all points in the interval November 27    2

Suppose that $y_1$ and $y_2$ are solutions to the homogenous DE $$y^{\prime \prime}+p(x)y^{\prime}+q(x)y=0$$ on $I$ and assume that for some $x_0 \in I$, we have $W(y_1,y_2)(x_0)>0$. Show that $W(y_1,y_2)(x)>0$ for all $x \in I$. I have no idea on h

## Solve $x^2u''+xu'-(x^2+\frac{1}{4})u=0$ using power series November 27

I stumbled upon this question in an old exam (I'm preparing for an exam of a course about ODEs). I didn't have much difficulty solving the Legendre and Hermite equations using power series, but this one is different.First of all, can you say this is

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