Problems 15 through 24 deal with ODEs and techniques discussed in Section \(4.8 .\) Classify the following ordinary differential equations by determining whether they are linear, what their order is, whether they are homogeneous, and whether their coefficients are constant. (a) \((\sin x) y^{\prime \prime}+\cos x=0\). (f) \(\frac{d y}{d t}=\frac{1}{1+y}\) (b) \(y^{\prime \prime}+y^{2}=2 y^{\prime}\). (g) \(\frac{d y}{d x}=\frac{1}{1+x}\). (c) \(\frac{d^{3} y}{d t^{3}}+\frac{2 d y}{d t}=\sin y\). (h) \(\frac{d^{5} y}{d x^{5}}=x^{6}+5 x+6\) (d) \(\frac{d}{d t}\left(y^{2}+2 y\right)=y\). (i) \(t \frac{d y}{d t}+t y=1\). (e) \(\frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+3 y=e^{t}+e^{-t}\).

Ordinary Differential Equations (ODEs) can be classified into linear and nonlinear based on the structure of the equation. If all the terms involving the dependent variable and its derivatives are linear, then the ODE is linear. This means that the dependent variable and its derivatives appear only to the first power and are not multiplied or divided by each other.
For example, the equation \(\frac{d^2 y}{d t^2} + 2 \frac{d y}{d t} + 3y = e^t + e^{-t}\) is linear because it fits the form y'' + a(t)y' + b(t)y = g(t), where a(t) and b(t) are functions of t, and g(t) is a nonzero function, making it inhomogeneous but still linear.

In contrast, if the dependent variable or its derivatives appear to any power other than one or if they are multiplied together, the ODE is nonlinear. For instance, the equation \y'' + y^2 = 2y'\ is nonlinear due to the y^2 term. Such nonlinearities often make solving the ODEs more complex.

The order of an ordinary differential equation is the highest derivative of the dependent variable that appears in the equation. Identifying the order is crucial as it determines the formulation and methods used for solving the ODE.
For example, the equation \(\frac{d^3 y}{d t^3} + 2 \frac{d y}{d t} = \text{sin} y\) is a third-order ODE because the highest derivative here is \( \frac{d^3 y}{d t^3}\).

First-order ODEs, like \( t \frac{d y}{d t} + t y = 1 \), involve only the first derivative \( \frac{d y}{d t} \). Second-order ODEs, such as \( (\text{sin} x) y'' + \text{cos} x = 0 \), include the second derivative \( y'' \). By classifying the order, we can apply the correct method to find the solution.

An ODE is homogeneous if every term is a function of the dependent variable and its derivatives, and the equation equals zero on the right-hand side. For example, \( (\text{sin} x) y'' + \text{cos} x = 0 \) is a homogeneous ODE because every term involves the dependent variable or its derivatives, and the right-hand side is zero.
Homogeneous ODEs can be simpler to solve as they often allow for eigenvalue methods and other proportional techniques.

In contrast, an inhomogeneous ODE has a nonzero right-hand side, typically involving terms that do not depend on the dependent variable or its derivatives. For example, \( y'' + 2y' + 3y = e^t + e^{-t} \) is inhomogeneous due to the \( e^t + e^{-t} \) term. Solving inhomogeneous ODEs often requires finding a particular solution to the nonhomogeneous part followed by solving the associated homogeneous equation.

Classification of an ODE can also depend on whether the coefficients of the terms involving the dependent variable and its derivatives are constant or variable.
**Constant Coefficients:** An ODE has constant coefficients if the coefficients are constants (numbers). For instance, \( y'' + 2y' + 3y = e^t + e^{-t} \) has constant coefficients (2 and 3). These ODEs are often simpler to solve, with solutions involving exponential functions, sine, and cosine functions.

**Variable Coefficients:** An ODE has variable coefficients if the coefficients are functions of the independent variable. For example, \( (\text{sin} x) y'' + \text{cos} x = 0 \) includes the coefficients \( \text{sin} x \) and \( \text{cos} x \), which are functions of x. Variable coefficient ODEs are typically more complex, and their solutions might require special methods or approximations, such as series solutions or numerical methods.