Final answer:
To solve the given initial-value problem using the substitution t = -x, differentiate twice with respect to t and replace the second derivatives in the original differential equation with the derivative in terms of x. Then, solve for y(x) using standard methods for linear homogeneous differential equations.
Step-by-step explanation:
To solve the given initial-value problem using the substitution t = -x, we first need to differentiate twice with respect to t. Let's start by finding the first derivative, using the chain rule:
dy/dt = dy/dx * dx/dt = dy/dx * (-1)
Next, we differentiate again:
d²y/dt² = d(dy/dt)/dt = d(dy/dx * (-1))/dt = -d²y/dx²
Now, replace the second derivative in the original differential equation with the derivative in terms of x:
4x²(-d²y/dx²) + y = 0
4x²d²y/dx² + y = 0
This is a linear homogeneous differential equation. Solve for y(x) using standard methods for this type of equation.