Question

To transform

xd^2y/dx^2+dy/dx=1/x into a linear differential equation with constant coefficients, the required substitution is

A. y=e^x
B. y=x^2
C. y= 1/x
​D. y=e^xlnx

Answer 1

The correct substitution to transform the given differential equation into a linear differential equation with constant coefficients is y = e^x. This substitution aligns with the characteristic behaviour of exponential functions and their derivatives.

Step-by-step explanation:

The student asks how to transform the given second-order non-linear differential equation into a linear differential equation with constant coefficients. To answer this question, we look for a substitution that simplifies the equation into the standard form of linear differential equation:

Ay'' + By' + Cy = D,
where A, B, C, and D are constants and y'' and y' are the second and first derivatives of y with respect to x, respectively.

Let's analyze the given differential equation:

xd^2y/dx^2 + dy/dx = 1/x.

To find the correct substitution, we note that derivatives of exponential and rational functions result in similar forms to their original function, which can considerably simplify the equation. Considering the answer choices, we rule out options that do not lead to linear equations with constant coefficients.

The correct substitution that would lead to a linear differential equation with constant coefficients is y = ex.