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blondinia [14]
2 years ago
7

3. Given the differential equation y"+y= 0 and the set of two functions sint-cost, sint + cost, show that this a fundamental set

of functions for the equation, and note the interval(s) over which this is true. wken (osxsihx | Cos sint
Mathematics
1 answer:
Harrizon [31]2 years ago
5 0

Answer with explanation:

Given the differential equation

y''+y=0

The two function let

y_1= sint -cost

y_2=sint+ cost

Differentiate y_1 and y_2

Then we get

y'_1= cost+sint

y'_2=cost-sint

Because \frac{\mathrm{d} sinx}{\mathrm{d} x} = cosx

\frac{\mathrm{d}cosx }{\mathrm{d}x}= -sinx

We find wronskin to prove that the function  is independent/ fundamental function.

w(x)=\begin{vmatrix} y_1&y_2\\y'_1&y'_2\end{vmatrix}

w(x)=\begin{vmatrix}sint-cost&sint+cost\\cost+sint&cost-sint\end{vmatrix}

w(x)=(sint-cost)(cost-sint)- (sint+cost)(cost+sint)

w(x)=sintcost-sin^2t-cos^2t+sintcost-sintcost-sin^2t-cos^2t-sintcost

w(x)=-sin^2t-cos^2t    

sin^2t+cos^2t=1

w(x)=-2\neq0

Hence, the given two function are fundamental set of function on R.

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