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Mathematics LibreTexts

1.2: Ordinary Differential Equations

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    $$E(v)=\int_a^bf(x,v(x),v'(x))\ dx$$

    and for given \(u_a,\ u_b\in{\mathbb R}\)

    $$V=\{v\in C^2[a,b]:\ v(a)=u_a,\ v(b)=u_b\},$$

    where \(y\) and \(f\) is sufficiently regular. One of the basic problems in the calculus of variation is

    (P) \(\min_{v\in V}E(v)\).

    Euler equation

    Let \(u\in V\) be a solution of (P), then


    in \((a,b)\).

    Exercise \(\PageIndex{1}\): Proof

    For fixed \(\phi\in C^2[a,b]\) with \(\phi(a)=\phi(b)=0\) and real \(\epsilon\), \(|\epsilon|<\epsilon_0\), set \(g(\epsilon)=E(u+\epsilon \phi)\). Since \(g(0)\le g(\epsilon)\) it follows \(g'(0)=0\). Integration by parts in the formula for \(g'(0)\) and the following basic lemma in the calculus of variations imply Euler's equation.

    Admissible Variations

    Figure Admissible Variations

    Basic lemma in the calculus of variations. Let \(h\in C(a,b)\) and

    $$\int_a^bh(x)\phi(x)\ dx=0$$

    for all \(\phi\in C_0^1(a,b)\). Then \(h(x)\equiv0\) on \((a,b)\).

    Proof. Assume \(h(x_0)>0\) for an \(x_0\in (a,b)\), then there is a \(\delta>0\) such that \((x_0-\delta,x_0+\delta)\subset(a,b)\) and \(h(x)\ge h(x_0)/2\) on \((x_0-\delta,x_0+\delta)\).

    \left(\delta^2-|x-x_0|^2\right)^2 & x\in(x_0-\delta,x_0+\delta)\\
    0 & x\in (a,b)\setminus[x_0-\delta,x_0+\delta]
    \end{array} \right. .

    Thus \(\phi\in C_0^1(a,b)\) and

    $$\int_a^b h(x)\phi(x)\ dx\ge \frac{h(x_0)}{2}\int_{x_0-\delta}^{x_0+\delta}\phi(x)\ dx>0,$$

    which is a contradiction to the assumption of the lemma.