(D^3+6D^2+12D+8)y=0

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Here is the video of differential equation-Higher Order Linear Differential Equations, where f(D)y=0 has solutions based on roots obtained by the auxiliary equation. Have patience and follow every step.
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Note :
Higher Order Linear Differential Equations(constant coefficients)
Def: Linear differential equation of order n:
An equation of the form
an(x)d^ny/dx^n+an-1(x)d^n-1/dx^n-1+.....+a1(x)dy/dx+a0(x)y=Q(x)
where a0, a1, a2,...,an and Q are continuous real functions in x defined on an intervel I is called a Linear Differential Equation of Order 'n' over the interval I.
Differential operator-notation:
Dy = dy/dx
D^2y = d^2y/dx^2
D^3y = d^3y/dx^3
.
.
.
.
D^ny = d^ny/dx^n
The polynomial in D
' D^n + P1 D^(n-1) + P2 D^(n-2) +....+ Pn '
is called a differential operator of order n and it is denoted by f(D).
where
f(D)≡D^n + P1 D^(n-1) + P2 D^(n-2) +....+ Pn
Auxiliary Equation:
f(m)=0
f(m)=0 may have real and distinct roots
The A.E of f(D)y=0 is f(m)=0
then
m^n + P1 m^(n-1) + P2 m^(n-2) +....+Pn =0
Let m1,m2,m3,m4 be equal roots.
Then
y=(c1+c2x+c3x^2+c4x^3)e^(m1x)
is the general solution of f(D)y=0.
Where c1,c2,c3,c4 are any real constants.
Order of a differential equation:
A differential equation is said to be of order n, if the nth derivative is the highest derivative in that equation.
Degree of a differential equation:
The highest power of the highest derivative.
For more such content👇
• Higher Order Linear Di...
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