Page 138 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 7: General Properties of Solutions of Linear Differential Equations of Order n

Therefore the general solution is Notes
y = C.F. + P.I.

3 3 1
/2
x
= e c 1 cos x c 2 sin x (2cos2x 3sin2 )
x
2 2 13
Self Assessment
11. Solve the following differential equations

(D D 2)y sin 2x

2
d y dy
12. 2 5 6y sin3x
dx dx
7.5 Summary

 The unit starts with the existence the uniqueness of the solution of nth order differential
equation.
 Here the nth order linear differential equation is reduced to a system of n first order
equations and the method of last unit applied.
 Some of the properties listed, help us in finding the general solution of the equation when
the coefficients are constant.

7.6 Keywords

Complementary functions are the solutions of the nth order differential equation without the
non-homogeneous term and involves n arbitrary constants.
Particular Integral (P.I.): It is the solution of non-homogeneous, nth order differential equation
without having any arbitrary constants.

7.7 Review Questions

1. Solve

2
d y dy
9 18 16y 0
dx 2 dx
2. Solve

4
d y
y 0
dx 4
3. Solve
4
3
2
(D D 9D 11D 4) y = 0
4. Solve
2
d y dy 4x
5 6y e
dx 2 dx

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