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To find the solution to the ODE, y(t), we need to use the inverse Laplace
transform, as follows:
OBJ ƒ ƒ Isolates right-hand side of last expression
ILAPµ Obtains the inverse Laplace transform
The result is
i.e.,
y(t) = -(1/7) sin 3x + y
o
cos √2x + (√2 (7y
1
+3)/14) sin √2x.
Check what the solution to the ODE would be if you use the function LDEC:
‘SIN(3*X)’ ` ‘X^2+2’ ` LDEC µ
The result is:
i.e., the same as before with cC0 = y0 and cC1 = y1.
Note: Using the two examples shown here, we can confirm what we
indicated earlier, i.e., that function ILAP uses Laplace transforms and inverse
transforms to solve linear ODEs given the right-hand side of the equation and
the characteristic equation of the corresponding homogeneous ODE.
Example 3
– Consider the equation
d
2
y/dt
2
+y = δ(t-3),
where δ(t) is Dirac’s delta function.
Using Laplace transforms, we can write:
L{d
2
y/dt
2
+y} = L{δ(t-3)},
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