Integration of homogeneous equations. Their general integral represents a system of similar curves. The equation (a + b x + c y) dx + (a’ + b’ x + c’ y) dy = c, may be rendered homogeneous. Particular case where the method fails. How the integration may be effected in such case.
Integration of the linear equation of the first order dy dx + P y = Q, where P and Q denote functions of x. Examples.
Remarks on the integration of equations of the first order which contain a higher power than the first of dy dx . Case in which it may be resolved in respect of dy dx . Case in which it may be resolved in respect of x or y.
Integrations of the equation y = x dy dx + φ ( dy dx ). Its general integral represents a system of straight lines. A particular solution represents the envelop of this system.
Solution of various problems in geometry which lead to differential equations of the first order.
Lessons 7–8. Integration of Differential Equations of Orders superior to the First.
The general integral of an equation of the m order contains m arbitrary constants.
(The demonstration is made to depend on the consideration of infinitely small quantities.)
Integration of the equation dmy dxm = φ (x.)
Integration of the equation d2y dx2 = φ (y, dy dx ).