# Initial Value Problem, First Order Linear Differential Equation

I am trying to solve an initial value problem but I am unsure if my result is acceptable. When I solve the differential equation two constants are introduced (C and A), but the solution can be written as the ratio of these constants.

Is it okay to consider this ratio as a single constant and use the initial condition to say C/A = -1/2 since u(1) = 1.

$$u' = \frac{u}{t} - t^{2}$$ has the solution $$u = -\frac{1}{2}t^{3} - \frac{Ct}{A}$$

A is the constant introduced by my integrating factor $$\mu(t) = e^{\int - \frac{1}{t}dt = A/t}, A = e^{c}$$ and C is the constant introduced when I perform the integral $$u = \frac{1}{\mu(t)} \int \mu(t) (-t^{2})dt$$

Essentially, is it ok to consider this to have only one constant to determine?

I hope this is clear enough,

thank you.

• but the quotient of $\frac{C}{A}$ is also a constant – Dr. Sonnhard Graubner Oct 18 '15 at 13:09
• This is really my point. I just want to be sure that it's fine to say something like $$b = \frac{C}{A}$$ and use the initial condition to determine the value of b. – jm22b Oct 18 '15 at 13:10
• You can always multiply the integration factor by a constant (not equal to zero), but this never simplifies the differential equation, so you can always set that constant to one for simplicity. – Kwin van der Veen Oct 18 '15 at 14:54