(Solved): Classify the order and if it is linear or non-linear It can be helpful to classify a differential eq ...

It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation \( - \) (what is the highest number of derivatives involved) and whether or not the equation is linear . Linearity is important because the structure of the the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear: 1. \( \frac{d y}{d t}+t y^{2}=0 \) 2. \( y^{\prime \prime}-y+y^{2}=0 \) 3. \( \left(1+y^{2}\right) \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+y=e^{t} \) 4. \( \frac{d^{2} y}{d t^{2}}+\sin (t+y)=\sin t \)