Yp(t) = A1 t^2 + A0 t + B0 t e(4t)
=> y ' = 2A1t + A0 + B0 [e^(4t) +4 te^(4t) ]
y ' = 2A1t + A0 + B0e^(4t) + 4B0 te^(4t)
=> y '' = 2A1 + 4B0e(4t) + 4B0 [ e^(4t) + 4te^(4t)
y '' = 2A1 + 4B0e^(4t) + 4B0e^(4t) + 16B0te^(4t)
Now substitute the values of y ' and y '' in the differential equation:
<span>y′′+αy′+βy=t+e^(4t)
</span> 2A1 + 4B0e^(4t) + 4B0e^(4t) + 16B0te^(4t) + α{2A1t + A0 + B0e^(4t) + 4B0 te^(4t) } + β{A1 t^2 + A0 t + B0 t e(4t)} = t + e^(4t)
Next, we equate coefficients
1) Constant terms of the left side = constant terms of the right side:
2A1+ 2αA0 = 0 ..... eq (1)
2) Coefficients of e^(4t) on both sides
8B0 + αB0 = 1 => B0 (8 + α) = 1 .... eq (2)
3) Coefficients on t
2αA1 + βA0 = 1 .... eq (3)
4) Coefficients on t^2
βA1 = 0 ....eq (4)
given that A1 ≠ 0 => β =0
5) terms on te^(4t)
16B0 + 4αB0 + βB0 = 0 => B0 (16 + 4α + β) = 0 ... eq (5)
Given that B0 ≠ 0 => 16 + 4α + β = 0
Use the value of β = 0 found previously
16 + 4α = 0 => α = - 16 / 4 = - 4.
Answer: α = - 4 and β = 0
Answer:
Explanation:
For free body diagram see attached sheet .
W is weight of steel girder acting at the middle point of its length . T is tension in the cable .
OB = √ ( 12² - 2² )
= 11.83 m .
OC = 11.83 / 2 = 5.915 m
Taking moment of tension T and weight W about point O
W x OC = T x OB
22 x 5.915 = T x 11.83
T = 22 x 5.915 / 11.83
= 11 kN
Considering forces acting in vertical direction and equating forces in opposite direction
T + R = W
R = W - T
= 22 - 11 = 11 KN
So force of grinder on the ground = R
= 11 KN.
Sound moves faster in warmer temperature because the particles move faster
The electric potential difference is the electric potential energy per unit charge
Explanation:
First of all, we define the concept of electric potential. The electric potential is a measure of the gradient of the electric field at a certain point of the space. The electric potential at a distance
from a positive charge of magnitude
is given by

where k is the Coulomb's constant.
Now we can define the electric potential energy and the electric potential difference:
- Electric potential energy is the energy possessed by a charge due to the presence of an electric field. For a charge of magnitude
immersed in an electric field, its potential energy is given by
, where V is the electric potential at the location of the charge. - The electric potential difference is simply the difference in electric potential between two points in the space. For instance, if the potential at point A is V(A) and the potential at point B is V(B), then the potential difference is

The electric potential energy is also defined as the work done on a charge q moved through a potential difference of
. Consequently, the potential difference
represents the work per unit charge done, i.e. the work done when moving a unitary charge through a potential difference
.
Learn more about potential difference and current:
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