Question

Use the method of Problem 11 to transform the given equation into the form \(\left[p(x) y^{\prime}\right]'+q(x) y=0\) $$ x y^{\prime \prime}+(1-x) y^{\prime}+\lambda y=0, \quad \text { Laguerre equation } $$

Short answer

Answer: The transformed equation is \((xy^{\prime})' + \lambda y = 0\), which is equivalent to the given differential equation \(xy^{\prime \prime} + (1 - x)y^{\prime} + \lambda y = 0\).

Step-by-step solution

Find p(x) and q(x)

Given the equation \(xy^{\prime \prime} + (1 - x)y^{\prime} + \lambda y = 0\), we observe that \(p(x) = x\) and \(q(x) = \lambda\).

Differentiate p(x) and q(x)

Now, we differentiate \(p(x) = x\) with respect to x. This gives us \(\frac{d}{dx}(p(x)) = \frac{d}{dx}(x) = 1\).

Find the transformed equation

Using Problem 11's method, transform the given equation by taking the derivative of \(p(x) y^{\prime}\): $$(p(x) y^{\prime})' = (xy^{\prime})' = x y^{\prime\prime} + y^{\prime}.$$ Now, we can plug in our values for \(p(x)\), \(q(x)\), and their derivatives into the required equation form \(\left[p(x) y^{\prime}\right]'+q(x) y=0\): $$(xy^{\prime})' + \lambda y = 0.$$ Comparing this new equation with the original equation, we can see that they are equivalent: $$xy^{\prime \prime} + (1 - x)y^{\prime} + \lambda y = (xy^{\prime})' + \lambda y = 0.$$ Hence, the given equation has been successfully transformed to the desired form.

Key concepts

Understanding Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. They are essential in modeling how systems change over time and can be found in various fields such as physics, engineering, and economics.
For instance, they might describe how heat moves through an object or how populations grow in an ecosystem.
The general idea is to find a function that satisfies the equation by relating the function to its derivatives.

• First-order differential equations involve the first derivative of the unknown function.
• Second-order and higher-order equations involve second or higher derivatives.

Understanding how to manipulate these equations is crucial for solving complex real-world problems. With practice, solving them becomes a valuable skill in scientific analysis.

The Role of Transformations

Transformations in differential equations involve changing the form of the equation to make it easier to solve. By transforming equations, we can sometimes reduce them to a simpler, standard form.
This process helps in recognizing patterns and applying known solutions.

• In the context of the exercise, we transformed the original equation into the form \([p(x) y^{\prime}]'+q(x) y=0\).
• By doing this, we align the equation with known forms, making it easier to apply mathematical tools to find solutions.

Such transformations often involve algebraic manipulation and differentiation.
They translate the problem into one where known solution methods can be effectively applied, facilitating the process of finding explicit solutions.

Second-Order Differential Equations

Second-order differential equations include the second derivative, represented as \(y^{\prime\prime}\). These equations can describe complex systems like oscillations or electromagnetic waves.
Solving these types of equations often requires more sophisticated techniques compared to first-order equations.

• The Laguerre equation, as given in the exercise, is an example of a second-order differential equation.
• It's typical for these equations to be associated with specific types of boundary conditions or constraints.

Solutions to second-order equations can be more intricate and may involve constants that need to be determined by the conditions of the problem.
Developing a solid understanding of second-order differential equations opens doors to analyzing more challenging scientific phenomena.