Mathematics: X^3y + 10x^2y + 16xy − 16y = 0; x, x^−4, x^−4 ln x, (0, [infinity]) Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x, x−4, x−4 ln x) = ≠ 0 for 0 x [infinity]. Form the general solution. y=

X^3y + 10x^2y + 16xy − 16y = 0; x, x^−4, x^−4 ln x,

ella3714

20.01.2022

Verifying that a solution satisifes the ODE is a matter of substituting the solution into the ODE. For example,

$y_1=x\implies{y_1}'=1\implies{y_1}''={y_1}'''=0$

$x^3{y_1}'''+10x^2{y_1}''+16x{y_1}'-16{y_1}=16x-16x=0$

The Wronskian for these solutions is

$W(x,x^{-4},x^{-4}\ln x)=\begin{vmatrix}x&x^{-4}&x^{-4}\ln x\\1&-4x^{-5}&x^{-5}(1-4\ln x)\\0&20x^{-6}&-x^{-6}(9-20\ln x)\end{vmatrix}=25x^{-10}$

which is non-zero for all $x\in(0,\infty)$ , so the solutions are indeed linearly independent.

The general solution is then

$y=C_1x+C_2x^{-4}+C_3x^{-4}\ln x$

0,0(0 оценок)