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Singular solution
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<p>A singular solution <i>ys</i>(<i>x</i>) of an <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a> is a solution that is <a href="/facts/Mathematical_singularity/Y5sMQY4I">singular</a> or one for which the <a href="/facts/Initial_value_problem/PqmJfGML">initial value problem</a> (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be <a href="/facts/Mathematical_singularity/Y5sMQY4I">singular functions</a>. </p><p>In some cases, the term <i>singular solution</i> is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as <a href="/facts/Tangent/sF0jH3EI">tangent</a> to every solution from a family of solutions. By <i>tangent</i> we mean that there is a point <i>x</i> where <i>ys</i>(<i>x</i>) = <i>yc</i>(<i>x</i>) and <i>y's</i>(<i>x</i>) = <i>y'c</i>(<i>x</i>) where <i>yc</i> is a solution in a family of solutions parameterized by <i>c</i>. This means that the singular solution is the <a href="/facts/Envelope_(mathematics)/jr5DkCGD">envelope</a> of the family of solutions. </p><p>Usually, singular solutions appear in differential equations when there is a need to divide in a term that might be equal to <a href="/facts/0_(number)/4NBRtKHc">zero</a>. Therefore, when one is solving a differential equation and using division one must check what happens if the term is equal to zero, and whether it leads to a singular solution. The <a href="/facts/Picard%25E2%2580%2593Lindel%25C3%25B6f_theorem/yKHSWoMJ">Picard–Lindelöf theorem</a>, which gives sufficient conditions for unique solutions to exist, can be used to rule out the existence of singular solutions. Other theorems, such as the <a href="/facts/Peano_existence_theorem/Y9obb40Q">Peano existence theorem</a>, give sufficient conditions for solutions to exist without necessarily being unique, which can allow for the existence of singular solutions. </p>