For differential equation y''-10y'+25y=0 give a basis and a solution space in term of basis.

phoreeldoefk

phoreeldoefk

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2022-08-19

For differential equation y''-10y'+25y=0 give a basis and a solution space in term of basis.

Answer & Explanation

Ean Hudson

Ean Hudson

Beginner2022-08-20Added 9 answers

The characteristic equation of the differential equation is:
r210r+25=0
Factorize the characteristic equation:
(r5)2=0
Determine the roots of the characteristic equation of the differential equation:
r=5
If the characteristic equation has two real roots r1 and r2, then the general solution is y(t)=c1er1t+c2er2t. If the characteristic equation has one real root r, then the general solution is y(t)=c1ert+c2tert. Since this characteristic equation has one real root, the general solution is:
y(t)=c1e5t+c2te5t
A basis for the solution space is then made up by the coefficients of the variables c1 and c2 in the general solution:
{e5t,te5t}
The solution space is then the span of the basis: Span {e5t,te5t}
Result:
Basis: {e5t,te5t}
Solution space: Span {e5t,te5t}

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