1. **Stating the problem:** We want to identify whether an ordinary differential equation (ODE) is linear or nonlinear and provide examples. 2. **Definition of linear ODE:** An ODE is linear if it can be written in the form $$a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x)$$ where the dependent variable $y$ and its derivatives appear to the first power and are not multiplied together. 3. **Important rules:** - The coefficients $a_i(x)$ depend only on the independent variable $x$. - No products or nonlinear functions (like $y^2$, $\sin(y)$) of $y$ or its derivatives appear. 4. **Example of a linear ODE:** $$\frac{dy}{dx} + 3y = \sin(x)$$ This is linear because $y$ and $\frac{dy}{dx}$ appear to the first power and are not multiplied. 5. **Example of a nonlinear ODE:** $$\frac{dy}{dx} = y^2 + x$$ This is nonlinear because of the $y^2$ term. 6. **Summary:** To identify linearity, check if $y$ and its derivatives appear only to the first power and are not multiplied or composed with nonlinear functions.