Question

Find values of m so that the function y = x is a solution of the given differential equation.

2y′′ + 9y′ − 7y = 0

Answer 1

$$y=x^m$$

$$y'=mx^{m-1}$$

$$y''=m(m-1)x^{m-2}$$

Given the differential equation

$$x^2y''+9xy'-7y=0$$

Substitute

$$m(m-1)x^{m-2}x^2+9mx^{m-1}x-7x^m=0$$

$$m^2-m+9m-7=0$$

$$m^2+8m-7=0$$

$$m=-4\pm\sqrt{(4)^2-1(-7)}$$

$$m_1=-4-\sqrt{23}, m_2=-4+\sqrt{23}$$