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 _______.
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
Important Questions from Differential Equations
What is the order of the differential equation ?
What is the degree of the differential equation ?
A solution of the differential equation
\(\left(\frac{d y}{d x}\right)^2-x \frac{d y}{d x}=0 \) is
If y = \(\rm\left(\frac{1}{x}\right)^x \), then value of \(\rm e^e\left(\frac{d^2 y}{d x^2}\right)_{x=e}\) is:
The general solution of the differential equation ydx - xdy = 0
