Question

Determine the order of the given differential equation; also state whether the equation is linear or nonlinear. $$ \left(1+y^{2}\right) \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+y=e^{\prime} $$

Short answer

Question: Determine the order and type (linear or nonlinear) of the given differential equation: \(\left(1+y^{2}\right)\frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}+y\) Answer: The order of the given differential equation is 2, and it is a nonlinear differential equation.

Step-by-step solution

Identify the highest order derivative in the equation

In the given differential equation, we can see that the highest derivative present is the second derivative, which is denoted as \(\frac{d^{2}y}{dt^{2}}\). Therefore, the order of this differential equation is 2.

Determine if the differential equation is linear or nonlinear

To determine if the equation is linear, we must check that all the terms in the equation represent a linear combination of the dependent variable and its derivatives. In a linear differential equation, the dependent variable and its derivatives are only raised to the power of one, and they are not multiplied or divided by each other or other nonlinear functions. Upon inspecting the given equation, we can see that it has the following terms: \(\left(1+y^{2}\right)\frac{d^{2}y}{dt^{2}}\), which has a nonlinear function \(y^2\) multiplied with a second derivative \(\frac{d^{2}y}{dt^{2}}\), \(t\frac{dy}{dt}\) is a linear term as t is multiplied by the first derivative of y, and finally, \(y\) is also a linear term because it represents the dependent variable without any nonlinear functions. So, as we have a nonlinear function multiplied with the second derivative in the given equation, this differential equation is nonlinear.

Conclusion

From our analysis, - The order of the given differential equation is 2 - The differential equation is nonlinear

Key concepts

Order of Differential Equation

Understanding the order of a differential equation is crucial as it signifies the highest derivative of the function present in the equation. In simple terms, the order gives us a hint about how many times you'll need to integrate to find the general solution. For instance, consider a differential equation displayed as \( \frac{d^{2} y}{d t^{2}} + t \frac{d y}{d t} + y = e^{t} \). Here, the highest derivative is the second derivative \( \frac{d^{2} y}{d t^{2}} \), which indicates that the order of the equation is 2. Why does this matter? The higher the order, the potentially more complex the solution becomes, as it involves additional integrations and constants to be determined. This detail also determines the number of initial conditions necessary to solve the equation uniquely, so it's a foundational aspect of solving differential equations.

From the exercise provided, by identifying the highest order derivative, the conclusion that the order is 2 forms the basis for understanding the rest of the equation's characteristics.

Linear vs Nonlinear Differential Equations

In the fascinating world of differential equations, the distinction between linear and nonlinear equations plays a vital role in dictating the method of solution and the complexity involved. Linear differential equations maintain a straightforward form, where the dependent variable \(y\) and its derivatives appear to the first power and are not multiplied or transformed by functions of \(y\). A linear equation might look like \( a_n(t) \frac{d^n y}{d t^n} + ... + a_1(t) \frac{d y}{d t} + a_0(t) y = g(t) \) with \(a_0(t), a_1(t), ..., a_n(t)\) being functions of \(t\) alone, and \(g(t)\) being the nonhomogeneous term.

On the flip side, nonlinear equations introduce terms where the dependent variable and its derivatives are taken to powers greater than one, or involve products of these derivatives, such as \(y\frac{d y}{d t}\) or \(\sin(y) \frac{d^2 y}{d t^2}\). The differential equation from the exercise \( (1+y^{2}) \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+y=e^{t} \) contains the term \(y^2\), which is multiplied by the second derivative, making the equation nonlinear. This distinction is imperative as linear equations can normally be solved with established analytical methods, whereas nonlinear equations may require more sophisticated or even numerical techniques.

Second Derivative

The second derivative, which we encounter in our exercise \( \frac{d^{2} y}{d t^{2}} \), is a profound tool that tells us about the rate at which the rate of change of a function is changing—it's the 'acceleration' of sorts for any function. It doesn't just predict future trends in the values of \(y\); it also provides insights into the concavity of the function's graph, allowing us to identify intervals where the function is curving upwards or downwards.

This understanding is particularly useful when solving differential equations, as it gives us a deeper insight into the behavior of solutions. For example, if \( \frac{d^{2} y}{d t^{2}} > 0 \) for all \(t\) in a particular interval, then the graph of \(y(t)\) is concave up in that interval. Similarly, if \( \frac{d^{2} y}{d t^{2}} < 0 \) over an interval, then \(y(t)\) is concave down. Knowing this can sometimes help in sketching the graph of the solution or in understanding the physical phenomena described by the equation.