How do u simplify 5/4 as fraction
2 answers:
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First check the characteristic solution:
<em>y''</em> + 4<em>y'</em> + 4<em>y</em> = 0
has characteristic equation
<em>r</em> ² + 4<em>r</em> + 4 = (<em>r</em> + 2)² = 0
with a double root at <em>r</em> = -2, so the characteristic solution is
For the particular solution corresponding to , we might first try the <em>ansatz</em>
but and
are already accounted for in the characteristic solution. So we instead use
which has derivatives
Substituting these into the left side of the ODE gives
so that 6<em>A</em> = 12 and 2<em>B</em> = 0, or <em>A</em> = 2 and <em>B</em> = 0.
For the second solution corresponding to , we use
with derivative
Substituting these gives
so that 4<em>C</em> = -8 and 4<em>C</em> + 4<em>D</em> = -12, or <em>C</em> = -2 and <em>D</em> = -1.
Then the general solution to the ODE is
Given the initial conditions <em>y</em> (0) = -2 and <em>y'</em> (0) = 1, we have
and so the particular solution satisfying these conditions is
or
You reflect a function with respect to the axis by changing its sign.
In fact, if you consider the two alternatives
and
you can see that you're choosing opposite outputs for the same inputs.
In other words, one graph will be made of points like , the other of points like
And the transformation
is exactly a reflection with respect to the axis.