find the general solution of the following first order differential equation.

t*(dy/dt) = (-y/t) + ((exp(1/t)/t^3)

could you show steps in working to solution.

To find the general solution of a first-order differential equation, you can follow these steps:

Step 1: Recognize the type of the equation.
The given equation is in the form t*(dy/dt) = (-y/t) + (exp(1/t)/t^3). This is a first-order linear ordinary differential equation.

Step 2: Write the equation in standard form.
To put the equation in standard form, we need to move all the terms to one side of the equation. Rearranging the terms, we get:
t*(dy/dt) + y/t = exp(1/t)/t^3

Step 3: Identify the integrating factor.
The integrating factor is given by the exponential of the integral of the coefficient of y with respect to t. In this case, the coefficient of y is 1/t. Therefore, the integrating factor (denoted by μ) can be found by integrating 1/t with respect to t:
μ = exp(∫(1/t) dt) = exp(ln|t|) = |t|

Step 4: Multiply the equation by the integrating factor.
Multiply both sides of the equation by the integrating factor (μ = |t|):
|t| * (t*(dy/dt) + y/t) = |t| * (exp(1/t)/t^3)

Simplifying the left side:
|t| * (t*(dy/dt) + y/t) = |t| * (t*(dy/dt)/dt + y/t^2) = |t| * (dy/dt + y/t^2)

Simplifying the right side:
|t| * (exp(1/t)/t^3) = (|t| * exp(1/t))/t^3

The equation becomes:
dy/dt + y/t^2 = (|t| * exp(1/t))/t^3

Step 5: Solve the resulting homogeneous equation.
Now we have a linear homogeneous differential equation of the form dy/dt + P(t)y = 0, where P(t) = 1/t^2. To solve this type of equation, we assume the solution y(t) has the form y(t) = e^(∫P(t) dt) * C, where C is the constant of integration.

Integrating P(t), we get: ∫(1/t^2) dt = -1/t

So the solution to the homogeneous equation is:
y(t) = C * e^(-1/t)

Step 6: Find a particular solution.
To find a particular solution, we need to use the method of variation of parameters or the method of undetermined coefficients. In this case, the latter method is easier.

Assume the particular solution as y(t) = A * exp(1/t) / t^3, where A is a constant to be determined.

Taking the derivative dy/dt, we have:
dy/dt = (A * exp(1/t) * ((-1/t^2) - (3/t^4)) + (A * exp(1/t) * (1/t^3))

Substituting these values back into the original equation, we get:
t * (dy/dt) = (-y/t) + (exp(1/t)/t^3)

Simplifying and collecting like terms, we have:
-(2A/t^4) = exp(1/t)/t^3

Comparing the powers of t on both sides, we get:
-2A = 1 => A = -1/2

So the particular solution is:
y(t) = (-1/2) * exp(1/t) / t^3

Step 7: Combine the general solutions.
The general solution of the differential equation is the sum of the homogeneous solution and the particular solution:

y(t) = C * exp(-1/t) + (-1/2) * exp(1/t) / t^3

Where C is a constant of integration.

This is the general solution to the given first-order differential equation.