Question

In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. In Problems 17 and 18 use Theorem 7.4.1 to reduce the given differential equation to a linear first-order DE in the transformed function $\mathbf{Y}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{=}\mathbf{L}\mathbf{\left\{}\mathbf{y}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right\}}$ Solve the first-order DE for Y(s) and then find $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{L}}^{\mathbf{-}\mathbf{1}}\mathbf{\left\{}\mathbf{Y}\mathbf{\right(}\mathbf{s}\mathbf{\left)}\mathbf{\right\}}$.

$\mathbf{2}{\mathbf{y}}^{\text{'}\text{'}}\mathbf{+}{\mathbf{ty}}^{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{10}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{y}}^{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Short answer

The first-order DE for$2y^{\text{'}\text{'}}+t{y}^{\text{'}}-2y=10$ is$y\left(t\right)=\frac{5}{2}{t}^2$

Step-by-step solution

Laplace Transform functions

Laplace Transform function is given below:

$L\left\{t^{n}f\left(t\right)\right\}=(-1{)}^n\frac{d^n}{d{s}^n}F\left(s\right)$

Apply the differential equation to a linear order differential equation.

Apply Integrating factor along with the differential equation