Question

Consider the differential equation:
\(\rm 2\dfrac{\mathrm{d^2y} }{\mathrm{d} t^2} +8y=0\) 
With initial conditions:
At t = 0, y = 0 and \(\dfrac{\mathrm{dy} }{\mathrm{d} t}=10\) . The value of y (up to two decimal places) at t = 1 is _______.

Answer 1

Concept:

For different roots of the auxiliary equation, the solution (complementary function) of the differential equation is as shown below.

Roots of Auxiliary Equation	Complementary Function
m1, m2, m3, … (real and different roots)	\({C_1}{e^{{m_1}x}} + {C_2}{e^{{m_2}x}} + {C_3}{e^{{m_3}x}} + \ldots\)
m1, m1, m3, … (two real and equal roots)	\(\left( {{C_1} + {C_2}x} \right){e^{{m_1}x}} + {C_3}{e^{{m_3}x}} + \ldots\)
m1, m1, m1, m4… (three real and equal roots)	\(\left( {{C_1} + {C_2}x + {C_3}{x^2}} \right){e^{{m_1}x}} + {C_4}{e^{{m_4}x}} + \ldots\)
α + i β, α – i β, m3, … (a pair of imaginary roots)	\({e^{α x}}\left( {{C_1}\cos β x + {C_2}\sin β x} \right) + {C_3}{e^{{m_3}x}} + \ldots\)
α ± i β, α ± i β, m5, … (two pairs of equal imaginary roots)	\({e^{α x}}\left( {\left( {{C_1} + {C_2}x} \right)\cos β x + \left( {{C_3} + {C_4}x} \right)\sin β x} \right) + {C_5}{e^{{m_5}x}} + \ldots\)

Calculation:

Given:

The auxiliary equation is

2λ² + 8 = 0

λ² = -4

The roots of auxiliary equation,

λ = +2i, -2i

Comparing the above roots with α + i β, α – i β, we get

α = 0, β = 1

∴ Complimentary function is 

y(t) = C₁ cos(2t) + C₂ sin(2t) .......(1)

y^'(t) = -2C₁ sin(2t) + 2C₂ cos (2t )........(2)

But we have,

y(0) = 0, y'(0) = 10

Putting above equations, we get

0 = C1 and 10 = 2C2

∴ C₁ = 0 and C₂ = 5

y(t) = 5 sin(2t)

y (t) = 5 sin(2) = 4.54