Question

Solve $$ 2 y \frac{d^{3} y}{d x^{3}}+2\left(y+3 \frac{d y}{d x}\right) \frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}=\sin x. $$

Short answer

Requires numerical methods due to its nonlinear nature.

Step-by-step solution

Recognize the Given Differential Equation

The given equation is:\[2 y \frac{d^3 y}{dx^3} + 2\left(y + 3 \frac{d y}{dx}\right) \frac{d^2 y}{dx^2} + 2 \left(\frac{d y}{dx}\right)^2 = \sin x.\]

Introduce Substitutions

Let \(z = \frac{d y}{dx}\). Then \(\frac{d y}{dx} = z\), \( \frac{d^2 y}{dx^2} = \frac{dz}{dx}\), and \(\frac{d^3 y}{dx^3} = \frac{d^2z}{dx^2}\). Substituting these into the original equation gives us:\[2 y \frac{d^2 z}{dx^2} + 2\left(y + 3z\right) \frac{dz}{dx} + 2z^2 = \sin x.\]

Simplify the Equation

Factor out the common terms:\[2\left(y \frac{d^2 z}{dx^2} + (y + 3z) \frac{dz}{dx} + z^2\right) = \sin x.\] Divide both sides by 2:\[y \frac{d^2 z}{dx^2} + (y + 3z) \frac{dz}{dx} + z^2 = \frac{1}{2}\sin x.\]

Recognize Pattern for Solver

Notice that the differential is in terms of functions of \(y\) and \(z\). To solve for \(y\), we need to find suitable methods or numerical tools depending on the problem's constraints. Since this is a highly nonlinear differential, numerical methods may be recommended.

Key concepts

higher-order derivatives

Higher-order derivatives extend the concept of differentiation to multiple levels. They provide deeper insights into the behavior of functions.

For example, the first derivative of a function represents its rate of change. The second derivative provides information about the curvature or concavity of the function.
Similarly, in the given differential equation, we encounter up to third-order derivatives:

\(2 y \frac{d^3 y}{dx^3} + 2\left(y + 3 \frac{d y}{dx}\right) \frac{d^2 y}{dx^2} + 2 \left(\frac{d y}{dx}\right)^2 = \sin x\)

Understanding these terms is crucial as they prove critical in analyzing complex systems in physics and engineering. Here is what each derivative represents:

• The term \( \frac{d^3 y}{dx^3} \) describes how the rate of curvature itself changes.
• The \( \frac{d^2 y}{dx^2} \) term shows the curvature or the acceleration if we consider a physical context.
• Finally, the \( \frac{d y}{dx} \) term gives the rate of change of the function itself.

When working on higher-order differential equations, you often deal with more complex behaviors and need sophisticated techniques to solve them.

substitution method

The substitution method is a powerful technique to simplify complex differential equations by introducing a new variable.

In our equation, we used the substitution:
\(z = \frac{d y}{dx}\)
This transformation helps us convert the original higher-order differential equation into a more manageable form by reducing the number of derivatives involved.

Once the substitutions are made, we get:
\(\frac{d y}{dx} = z\)
\( \frac{d^2 y}{dx^2} = \frac{dz}{dx}\)
\( \frac{d^3 y}{dx^3} = \frac{d^2 z}{dx^2}\)

By substituting these into the original equation:
\(2 y \frac{d^2 z}{dx^2} + 2\left(y + 3z\right) \frac{dz}{dx} + 2z^2 = \sin x\)

We can further simplify and solve the equation using additional methods or numerical approaches. Substitution saves time and reduces the complexity of the problem, making it an indispensable tool in solving differential equations.

numerical methods

Numerical methods are essential when analytical solutions are complicated or impossible to obtain.
They use algorithms to provide approximate solutions to differential equations.

Common numerical methods include:

• **Euler's Method**: One of the simplest, it uses linear approximations of the function and solves step-by-step
  
• **Runge-Kutta Methods**: These offer more accuracy by considering multiple points within each interval
• **Finite Difference Methods**: These convert differential equations into algebraic equations that can be solved iteratively

Given that our original equation is nonlinear and highly complex:
\(2 y \frac{d^3 y}{dx^3} + 2\left(y + 3 \frac{d y}{dx}\right) \frac{d^2 y}{dx^2} + 2 \left(\frac{d y}{dx}\right)^2 = \sin x\)

Numerical methods like Runge-Kutta can provide a useful way to approximate the solution.
Software tools such as MATLAB, Mathematica, or even Python's SciPy library are often employed to handle these computations efficiently.

By breaking down the steps involved, numerical methods iterate towards a solution, making them invaluable for solving real-world problems where analytical methods fall short.