How this is reduced to an equation of the first order. Solution of various problems in geometry which conduct to differential equations of the second order.
Lessons 9–10. On Linear Equations.
When a linear equation of the mth order contains no term independent of the unknown function and its derivatives, the sum of any number whatever of particular integrals multiplied by arbitrary constants is also an integral. From this the conclusion is drawn that the general integral of this equation is deducible from the knowledge of m particular integrals.
Application to linear equations with constant co-efficients. Their integration is made to depend on the resolution of an algebraical equation. Case where this equation has imaginary roots. Case where it has equal roots. The general integral of a linear equation of any order, which contains a term independent of the function, may be reduced by the aid of quadratures to the integration of the same equation with this term omitted.
Lesson 11. Simultaneous Equations.
General considerations on the integration of simultaneous equations. It may be made to depend on the integrations of a single differential equation. Integration of a system of two simultaneous linear equations of the first order.
Lesson 12. Integrations of Equations by Series.
Development of the unknown function of the variable x according to the powers of x-a. In certain cases only a particular integral is obtained. If the equation is linear, the general integral may be deduced from it by the variation of constants.
Lessons 13–16. Partial Differential Equations.
Elimination of the arbitrary functions which enter into an equation by means of partial derivatives. Integration of an equation of partial differences with two independent variables, in the case where it is linear in respect to the derivatives of the unknown function. The general integral contains an arbitrary function.