90 Example of Laplace transform of differential equation

To find the Laplace transform of

$$y^{\prime \prime \prime }+2y^{\prime \prime }-y^{\prime }+y=u$$

$$y^{\prime \prime }(0)=5,y^{\prime }(0)=-1,y(0)=7$$

where $u\in (E)$.

Solution. The Laplace transforms are found recursively, with

$$ℒ(y)(s)=\widehat{y}(s)$$

$$ℒ(y^{\prime })(s)=s\widehat{y}(s)-7$$

$$ℒ(y^{\prime \prime })(s)=s^2\widehat{y}(s)-7s+1$$

$$ℒ(y^{\prime \prime \prime })(s)=s^3\widehat{y}(s)-7s^2+s-5$$