- #1
$u(x) = e^{\int (2/t) dt}= e^{2\ln t} = \left(e^{\ln t}\right)^2 = t^2$.karush said:#18
$ ty'+2y=\sin t \quad y(\pi/2)=1 \quad t\ge 0$
$y'+\dfrac{2}{t}y=\dfrac{\sin t}{t}$
so
$u(x) = e^{\int 2/t dt}= e^{t^2/4}$
Opalg said:$u(x) = e^{\int (2/t) dt}= e^{2\ln t} = \left(e^{\ln t}\right)^2 = t^2$.
Yes, keep going! (Yes) (Integrate both sides with respect to $t$.)karush said:$t^2 y'+\dfrac{2t^2}{t}y=\dfrac{t^2\sin t}{t}$
$t^2 y'+2ty=t\sin t$
$(yt^2)'=t\sin t$
proceed ?
$(yt^2)'=t\sin t$Opalg said:Yes, keep going! (Yes) (Integrate both sides with respect to $t$.)
skeeter said:$y = -\dfrac{\cos{t}}{t} + \dfrac{\sin{t}}{t^2} + \dfrac{C}{t^2}$
$y\left(\dfrac{\pi}{2}\right) = 1$
$1 = 0 + \dfrac{4}{\pi^2} + \dfrac{4C}{\pi^2}$
solve for $C$
karush said:$+ \dfrac{\pi^2-4}{4t^2}$
Where did this come from?
The equation represents a differential equation, where the derivative of y (represented by y') is multiplied by a constant (17) and then subtracted from 2 times y. The equation also includes an initial condition, where the value of y at t=0 is equal to 2.
Differential equations are used to model real-world phenomena that involve rates of change. In this case, the equation represents a relationship between the rate of change of y and the function e^{2t}. Solving this equation can help us understand how y changes over time.
There are various methods for solving differential equations, such as separation of variables, integrating factors, and substitution. In this case, we can use the integrating factor method to solve for y.
The initial condition represents the starting point for the solution of the differential equation. In this case, it tells us that when t=0, the value of y is equal to 2. This helps us determine the specific solution to the equation.
Yes, this equation can be used to model many real-world situations, such as population growth, radioactive decay, and chemical reactions. The specific interpretation of the equation would depend on the context in which it is used.