General solution

Let y = e^mx be a trial solution.

so dydx = me^mx & d²ydx² = m²e^mx

now plugging in the values of dy/dx and d²y/dx² in the given equation we get...

100 m²e^mx - 20 me^mx + e^mx = 0

so e^mx( 100 m² - 20m + 1) = 0

e^mx ≠0

so 100 m² - 20m + 1 = 0

or ( 10m - 1)² = 0

so m = 110 , 110

i.e. the roots of the above quadratic are real and equal. ^**
so general solution of the given differential equation is :

y = ( A+ Bx)^(1/10)x

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^** a little info....

for the general equation m² + P₁ m + P₂ = 0 (called the auxillary equation)

let m₁ and m₂ be the roots...

CASE I

when roots are REAL AND UNEQUAL

the general solution is:

y = Ae^m₁x + Be^m₂x

where A and B are two independent arbitary constants.

CASE II

when roots are REAL AND EQUAL

The general solution is :

y = (A + Bx)e^mx

where A and B are two independent arbitary constants.

and m = m₁ = m₂ (in other words m is the equal root)

CASE III

when roots are CONJUGATE COMPLEX QUANTITIES

the general solution is y = Ae^(alpha)xcos{((beta)x) + B}

where A and B are two independent arbitary constants

and m₁ = (alpha) + i (beta)

and m₂ = (alpha) - i (beta)
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