Solve the for the general solution of the differential equation. (D^{2}+2D-1)y=0

melodykap

melodykap

Answered question

2021-04-09

Solve the for the general solution of the differential equation.
(D2+2D1)y=0

Answer & Explanation

hajavaF

hajavaF

Skilled2021-04-11Added 90 answers

Step 1
We have to solve the differential equation:
(D2+2D1)y=0
The given equation is in symbolic form
where,
D=dydx
D2=d2ydx2
Its auxiliary equation would be:
D2+2D1=0
Solving using the formula of quadratic equation,
here,
a=1
b=2
c=-1
Therefore,
D=b±b24ac2a
=2±224(1)(1)2×1
=2±4+42
=2±82
=2±222
=2(1+±2)2
=1±2
=1+2,12
Step 2
Hence, there are two solution of auxiliary equation
a=1+2
b=12
We know solution for this type of differential equation is given as:
y=c1eax+c2ebx
where, c1andc2 are arbitrary constants.
Putting values of a and b,
y=c1eax+c2ebx
y=c1e(1+2)x+c2e(12)x
Hence, solution of given differential equation is y=c1e(1+2)x+c2e(12)x.

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