Question

Solve following differential equation by changing independent variable: `(d^(2)y)/(dx^(2))=-(1)/(x)(dy)/(dx)+4x^(2)y=x^(4)`

Answer 1

Answer: Let u = ln(x). Then, we have:

y = y(u)

dy/dx = dy/du * du/dx = dy/du * (1/x)

d^2y/dx^2 = d/dx (dy/du * du/dx) = d^2y/du^2 * (1/x)^2 - dy/du * (1/x^2)

Substituting these into the original differential equation yields:

d^2y/du^2 - dy/du + (4 - e^u)y = e^(2u)

This is now a second-order ordinary differential equation in y with constant coefficients, which can be solved using standard methods such as the characteristic equation or undetermined coefficients.

Explanation: