Solve the given initial value problem using the method of Laplace transforms. y'' + 7y' + +10y=tu(t-3); y(0) = 0, y'(0) = 3

Solve using Laplace transforms y''+2y'-6y=7e-3t, where y(0)=1, and y'(0)=-2

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

Use Laplace transforms to solve the initial value problem. x''+4x'+3x=1;  x(0)=0=x'(0)