q are the order and degree of the differential equation y dy d y + when ( ) p ( ) p ( ) p > ( ) p exists and q does not exist . The solution of the differential equation dy is ( ) y sin ( ) x sin ( ) y ( ) x . The solution of the differential equation dy = is ( ) y Ce x ( ) y ( ) y Ce ( ) y . The general solution of the differential equation log dy = is ( ) e ( ) e ( ) e − + ( ) e .
The solution of dy y x is ( ) ( ) ( ) ( ) x . The solution of the differential equation dy φ φ ' is ( ) x k φ = ( ) φ kx = ( ) y k φ = ( ) φ ky = . If sin x is the integrating factor of the linear differential equation dy Py Q , then P is ( ) logsin x ( ) cos x ( ) tan x ( ) cot x . The number of arbitrary constants in the general solutions of order n and n + are respectively ( ) n − , ( ) n + ( ) n ( ) n + , .
The number of arbitrary constants in the particular solution of a differential equation of third order is ( ) ( ) ( ) ( ) - - Ordinary Differential Equations . Integrating factor of the differential equation dy is ( ) x + ( ) x + ( ) x + ( ) x + . The population P in any year t is such that the rate of increase in the population is proportional to the population. Then ( ) P Ce kt ( ) P Ce kt ( ) P Ckt ( ) P .
P is the amount of certain substance left in after time t . If the rate of evaporation of the substance is proportional to the amount remaining, then ( ) P Ce kt ( ) P Ce kt ( )