# Toronto Math Forum

## MAT244--2018F => MAT244--Lectures & Home Assignments => Topic started by: Qinger Zhang on September 23, 2018, 04:50:56 PM

Title: problem 6 in 1.3
Post by: Qinger Zhang on September 23, 2018, 04:50:56 PM
why (d^3y)/(dt^3) + t (dy)/(dt) +(cos^2(t))y = t^3 is linear? It doesn't have second derivative of dy dt, so it is not a linear equation.
Title: Re: problem 6 in 1.3
Post by: Wei Cui on September 23, 2018, 05:56:09 PM
A linear differential equation is in the form of $a_0(t)y^{(n)} + a_1(t)y^{(n-1)}+ ... + a_n(t)y = g(t)$, and also $a_i(t)$ can be a constant function which is: $a_i(t) = C$.

In this case, $a_1(t) = 0$, so the second derivative $y^{''}$ disappears. However, it's still a linear differential equation.