Ordinary Differential Equations MA26600 Spring 2026

Uniqueness Theorem for First-order Linear IVP’s.

Remarks.

1. Compare this theorem to the Existence and Uniqueness Theorem we learned in Section 1.3.

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   Theorem 38. Existence and Uniqueness for First-order IVP’s.

   If both \(f(x,y)\) and \(D_y f(x,y)\) are continuous on some rectangle \(R\) that contains the point \((a,b)\) in the interior, then there is an interval \(I\) containing \(a\) for which the initial value problem

   \begin{equation*} \frac{dy}{dx}=f(x,y), \qquad y(a)=b \end{equation*}

   has a unique solution on the interval \(I\text{.}\)

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2. First-order linear differential equations do not have singular solutions. All of the solutions are of the form given in (✶).

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3. You can slightly modify our definition of the integrating factor for the IVP so that \(C\) can be found automatically. It is easy to check that \(\rho(x_0)=1\) and \(y(x_0)=y_0\) with the following definitions.

   \begin{gather} \rho(x)=e^{\int_{x_0}^x P(w) \, dw}\tag{†}\\ y(x)= \frac{1}{\rho(x)} \Biggr[\int_{x_0}^x \rho(w)Q(w) \, dw +y_0 \Biggr]\tag{#} \end{gather}

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Mixture Problems.