FIRST ORDER DIFFERENTIAL EQUATIONS
Equations Reducible to Exact Form.
4. Solutions by Substitutions
2.4.2 Bernoulli’s Equation
In this chapter we describe procedures for solving 4 types of differential equations of first order, namely, the class of differential equations of first order where variables x and y can be separated, the class of exact equations (equation (2.3) is to be satisfied by the coefficients of dx and dy, the class of linear differential equations having a standard form (2.7) and the class of those differential equations of first order which can be reduced to separable differential equations or standard linear form by appropriate. 2.1 Separable Variables
Definition 2.1: A first order differential equation of the form
is called separable or to have separable variables.
Method or Procedure for solving separable differential equations
(i) If h(y) = 1, then
dy = g(x) dx
Integrating both sides we get
where c is the constant of integral
We can write
where G(x) is an anti-derivative (indefinite integral) of g(x) (ii) Let [pic]
that is f(x,y) can be written as the product of two functions, one function of variable x and other of y. Equation
can be written as
By integrating both sides we get
where H(y) and G(x) are anti-derivatives of [pic]and [pic], respectively.
Example 2.1: Solve the differential equation
Solution: Here [pic], [pic]and [pic]
H(y) = G(x) + C
or lny = lnx + lnc
(See Appendix )
lny – lnx = lnc
y = cx
Example 2.2: Solve the initial-value problem...
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