Use variation of parameters to find a general solution to the differential equation given that the functions and Y₁ Y₂ are linearly independent solutions to the corresponding homogeneous equation for t> 0. ty" - (t+1)y'+y=t²; Y₁ = ¹, y₂=t+1

Consider the differential equation x²y" - 9xy' + 24y = 0; x4, x6, (0, ∞). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. w(x4, x) = The functions satisfy the differential equation and are linearly independent since # 0 Form the general solution. y = for 0 < x < 0,

An integrating factor for the linear differential equation Y- 2y : = 12x is u= | A. 2xe2x В. — 2 C.-2x D. In 2x O E. e Е. -2x

Consider the function of two variables  u (x,t) = 2e-t sin (2x)  Select the option that gives a partial differential equation for which this function of two variables is a solution.  A. (∂2u/∂x2) = 4(∂2u/∂t2)  B. (∂2u/∂x2) = (1/4)(∂2u/∂t2)  C. (∂2u/∂x2) = - (1/4)(∂2u/∂t2)  D. (∂2u/∂x2) = -4 (∂u/∂t)  E. (∂2u/∂x2) = 4 (∂u/∂t)