dx2d2y+4dxdy+13y=0;y(0)=1,y′(0)=4
The given equation can be written as
(D2+4D+13)y=0;y(0)=1,y′(0)=4.
The auxiliary equation is,
m2+4m+13=0
Solving for m,
m=2−4±16−4∗13 =2−4±−36 =−2±3i.
The solution is,
y(x)=e−2x(Acos3x+Bsin3x),
where A and B are constants.
To find A and B, we use the given initial conditions
y(0)=1, y'(0)=4.
Now,
y′(x)=e−2x(−3Asin3x+3Bcos3x) −2e−2x(Acos3x+Bsin3x) =e−2x(−3Asin3x+3Bcos3x)−2y(x).
Using the condition
y(0) = 1, in y(x) we get A = 1.
Using the condition
y(0) = 1, y'(0) = 4 and A = 1 in y'(x), we get
y'(0) = (0 + 3B) - 2y(0)
4 = 3B - 2
B = 2.
Thus, the solution is
y(x)=e−2x(cos3x+2sin3x).
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