Methods of variation of parameters- advance engineering mathe mathematics

mathod of variation of parameters

1. Method of Variation of Parameters By Kaushal Patel

2. Method of Variation of Parameters Langrage invented the method of variation of parameters. Consider differential equation of the form f(D)y=X. when X is of the form or any function of x, then the shortcut methods are available which will discuss later on. If X be of any other form say then we have to use one of the following methods. The method of partial fractions The method of variation of parameters  

3. Continue… Consider the second order linear differential equation with constant co-efficient ……( i ) Let the complementary function of ( i ) be Then and satisfy the equation ……(ii)  

4. Continue… Let us assume that the particular integral of ( i ) be ……(iii) where u and v are unknown functions of x Differentiating w.r. to x, we have ……(iv) To determine two unknown functions u and v, we need two equations. We assume that ……(v) (iv) reduces to ……(vi)  

5. Continue… Differentiating w.r. to x, we get Substituting the values of and in ( i ), we have ……(vii) But and satisfy equation (ii) and Equation (viii) takes the form, ……(viii)  

6. Continue… Solving (v) and (ix), we get and Integrating, we get and , where Substituting the values of u and v in (iii), we have + ……(ix)  

7. Example Find general solution of by method of variation of parameters. Solution:- The given differential equation can be written as, The auxiliary equation is, =>  

8. Continue… Hence, complimentary function is Now we take =3 Particular integral is,  

9. Continue… Hence, general solution is y(x)