Use the Laplace transform to solve the given initial-value problem. у" - 10y' + 25y 3 t, y(0) %3D 0, У(0) - 1 y(t) =

Use the Laplace transform to solve the following initial value problem: y''+8y'=0, y(0)=-2, y'(0)=-2

Use Laplace transforms to solve the following initial value problem. x" - 4x = 5t; x(0) = x'(0) = 0 Click the icon to view the table of Laplace transforms.

An equation of the form +2d²y dt2 dy +at dt + By = 0, t> 0, where a and ẞ are real constants, is called an Euler equation. α (a). Let x = Int and calculate dy/dt and d²y/dt² in terms of dy/dx and d²y/dx². (b) Show that one can use the results of part (a) to transform the original equation into d²y dy + (α − 1). dx² + By = 0. dx Observe now that the resulting differential equation has constant coefficients. (c) Show that if y₁(x) and y2(x) form a fundamental set of solutions of the latter equation in part (b), then y₁ (Int) and y2 (Int) form a fundamental set of solutions of the original equation. (d) Using all above observations to solve 1²y" + 4ty' + 2y = 0