Question

Q: 5 Solve the initial value problems :
a) y" + 4y'+ 4y = 0
y(0) = 1 and y'(0) = -1 .

b) y"-2y'-3y=0.
y(0) =1 and y'(0) = 3


See Question 6 from Pic :


#####No Improper Answer ######

Answer 1

Final Result :
And :5 1)
$y = (x + 1) {e}^{ - 2x}$

2)
$y = {e}^{3x}$

Ans 6 :

1) y(x) = Ax^3 + B/x^4
2) y(x) = 1/x^2(Acos (3ln x) + B sin (3 ln x))
3) y(x) = (A+B ln x) x

Homogeneous 2nd Linear equation with constant Co - efficients
------------------------------------------------------------------

=>If the ODE is of the form

ay"+by'+cy=0. where x belongs to some interval.
and a, b, c are constants then two independent solutions (I. e. basis)
depend on the quadratic equation

$a {m}^{2} + bm + c = 0$
For Answer 5
See Pic 1 ! (both parts)

Use initial condition to find the value of constants in general solution.

6) See Pic for calculation :

Here, we will convert Euler's Cauchy Equation into Homogeneous 2nd Order Linear equation
with constant Co efficient.

By substituting x = e^t and
y(x) = y(e^t)= u(t)