Differential Equations
CHAPTER 2
FIRST ORDER DIFFERENTIAL EQUATIONS
2.1 Separable Variables
2.2 Exact Equations 2.2.1 Equations Reducible to Exact Form.
2.3 Linear Equations 4. Solutions by Substitutions 2.4.1 Homogenous Equations 2.4.2 Bernoulli’s Equation
2.5 Exercises 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
[pic] is called separable or to have separable variables.
Method or Procedure for solving separable differential equations
(i) If h(y) = 1, then [pic] or dy = g(x) dx Integrating both sides we get [pic] or [pic] where c is the constant of integral We can write [pic] where G(x) is an anti-derivative (indefinite integral) of g(x)
(ii) Let [pic] where [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 [pic] can be written as [pic] By integrating both sides we get [pic] where [pic] or [pic] where H(y) and G(x) are anti-derivatives of [pic]and [pic], respectively.
Example 2.1: Solve the differential equation
[pic]
Solution: Here [pic], [pic]and [pic]
[pic], [pic]
Hence
H(y) = G(x) + C
or lny = lnx + lnc (See Appendix ) lny – lnx = lnc [pic] [pic] y = cx
Example 2.2: Solve the initial-value problem
FIRST ORDER DIFFERENTIAL EQUATIONS
2.1 Separable Variables
2.2 Exact Equations 2.2.1 Equations Reducible to Exact Form.
2.3 Linear Equations 4. Solutions by Substitutions 2.4.1 Homogenous Equations 2.4.2 Bernoulli’s Equation
2.5 Exercises 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
[pic] is called separable or to have separable variables.
Method or Procedure for solving separable differential equations
(i) If h(y) = 1, then [pic] or dy = g(x) dx Integrating both sides we get [pic] or [pic] where c is the constant of integral We can write [pic] where G(x) is an anti-derivative (indefinite integral) of g(x)
(ii) Let [pic] where [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 [pic] can be written as [pic] By integrating both sides we get [pic] where [pic] or [pic] where H(y) and G(x) are anti-derivatives of [pic]and [pic], respectively.
Example 2.1: Solve the differential equation
[pic]
Solution: Here [pic], [pic]and [pic]
[pic], [pic]
Hence
H(y) = G(x) + C
or lny = lnx + lnc (See Appendix ) lny – lnx = lnc [pic] [pic] y = cx
Example 2.2: Solve the initial-value problem