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2. Reduction of order - Wikipedia

   en.wikipedia.org/wiki/Reduction_of_order

   Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution () is known and a second linearly independent solution () is desired. The method also applies to n-th order equations. In this case the ansatz will yield an (n−1)-th ...

3. Second derivative - Wikipedia

   en.wikipedia.org/wiki/Second_derivative

   The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.

4. Differential equation - Wikipedia

   en.wikipedia.org/wiki/Differential_equation

   The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.

5. Abel's identity - Wikipedia

   en.wikipedia.org/wiki/Abel's_identity

   In mathematics, Abel's identity (also called Abel's formula [1] or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation.

6. Separation of variables - Wikipedia

   en.wikipedia.org/wiki/Separation_of_variables

   This equation is an equation only of y'' and y', meaning it is reducible to the general form described above and is, therefore, separable. Since it is a second-order separable equation, collect all x variables on one side and all y' variables on the other to get: (′) (′) =.

7. Symmetry of second derivatives - Wikipedia

   en.wikipedia.org/wiki/Symmetry_of_second_derivatives

   It can matter, looking at the first-order terms, which is applied first. This leads to the construction of pathological examples in which second derivatives are non-symmetric. This kind of example belongs to the theory of real analysis where the pointwise value of functions matters.