Using Laplace transforms find the solution to a differential equation.

Question

$$ y'+\frac{2}{t}y= t - 1 + \frac{1}{t}. $$

with initial condition $y(1)=1/2$

I know how to do it using Laplace transformations when initial condition is of $y(0)=a$ where $a$ is a constant but not otherwise because we were taught the Laplace transform of $y'$ is $sY(s)-y(0).$

I can solve this equation easily with a linear approach.

solution being

$$y(t)=(t^2/4)-(t/3)+(1/2)+(1/(12t^2))$$

Can someone show me how to solve this initial value problem using Laplace transforms? I'm stuck.

Answer 1

Hint:

Since you know how to solve it using LT (I do not think it is possible to do so, but solving it using an Integrating Factor or as an Exact Equation is doable), proceed and solve it that way while maintaining a constant for $y(0) = c$ in all the calculations.

Once you have the solution for $y(t)$, substitute $y(1) = \dfrac{1}{2}$ to solve for $c$.

The solution should be:

$$y(t) = \dfrac{1}{12}\left(3 t^2 + \dfrac{1}{t^2} - 4t + 6\right)$$