Question

A nonhomogeneous Euler equation \(t^{2} y^{\prime \prime}+\alpha t y^{\prime}+\beta y=g(t)\) is known to have the given general solution. Determine the constants \(\alpha\) and \(\beta\) and the function \(g(t)\). $$ y(t)=c_{1} t^{2}+c_{2} t^{-1}+2 t+1, \quad t>0 $$

Short answer

Question: Determine the constants α, β, and the function g(t) for the given nonhomogeneous Euler equation and its general solution. Euler equation: \(t^{2}y'' + \alpha ty' + \beta y = g(t)\) General solution: \(y(t) = c_1 t^2 + c_2 t^{-1} + 2t + 1\) Answer: α = 1, β = 1, and g(t) = \(4c_1t^2 - 5c_2t^{-1} + 3t + 1\).

Step-by-step solution

Compute the first and second derivatives of the general solution

First, we need to compute the first and second derivatives of the general solution \(y(t)=c_{1} t^{2}+c_{2} t^{-1}+2 t+1\). Using the power rule for differentiation, we get: $$ y'(t) = \frac{d}{dt}\left(c_{1}t^{2}+c_{2}t^{-1}+2t+1\right) = 2c_{1}t + \left(-c_{2}t^{-2}\right) + 2, $$ and $$ y''(t) = \frac{d^{2}}{dt^{2}}\left(c_{1}t^{2}+c_{2}t^{-1}+2t+1\right) = 2c_{1} - 2c_{2}t^{-3}. $$

Substitute the derivatives into the Euler equation

Next, we substitute the derivatives of the general solution into the Euler equation: $$ t^{2} y^{\prime\prime}(t) + \alpha t y^{\prime}(t) + \beta y(t) = g(t). $$ Substituting the first and second derivatives, we get: $$ t^{2}(2c_{1} - 2c_{2}t^{-3}) + \alpha t (2c_{1}t - c_{2}t^{-2} + 2) + \beta(c_{1}t^{2} + c_{2}t^{-1} + 2t + 1) = g(t). $$

Identify the constants \(\alpha\) and \(\beta\)

Now, let's simplify the expression and gather terms: $$ (2\beta c_{1} + 2c_{1})t^{2} + (\alpha c_{2} - 6\beta c_{2})t^{-1} + (\alpha + 2\beta)t + \beta = g(t). $$ Comparing with the given Euler equation, we conclude that: $$ \alpha = 1, $$ and $$ \beta = 1. $$

Find the function \(g(t)\)

Finally, using the values of \(\alpha\) and \(\beta\) found in Step 3, we can write the function \(g(t)\): $$ g(t) = (2\beta c_{1} + 2c_{1})t^{2} + (\alpha c_{2} - 6\beta c_{2})t^{-1} + (\alpha + 2\beta)t + \beta. $$ Substitute \(\alpha = 1\) and \(\beta = 1\) into the equation and simplify: $$ g(t) = 4c_{1}t^{2} - 5c_{2}t^{-1} + 3t + 1. $$ Thus, the constants \(\alpha\) and \(\beta\) and the function \(g(t)\) are given by: $$ \alpha = 1, \quad \beta = 1, \quad \text{and} \quad g(t) = 4c_{1}t^{2} - 5c_{2}t^{-1} + 3t + 1. $$

Key concepts

Differential Equations

Differential equations are equations that involve a function along with its derivatives. These equations describe various phenomena such as heat, light, and fluid dynamics by relating functions to their rates of change. In simple terms, they function as mathematical tools to model real-world situations. Differentiating helps in determining how a quantity changes, reflected in the predictable patterns described by equations. When solving differential equations, we often look for relationships that fulfill the conditions expressed, covering various branches such as ordinary and partial differential equations.
Ordinarily, differential equations relate functions of one variable, while their solutions are essential to predicting behaviors in engineering, physics, and beyond.

General Solution

To solve nonhomogeneous differential equations, such as the Euler equation we have here, solutions are structured into general and particular solutions. The general solution includes the complementary function and a specific part that satisfies the equation. When we discuss the general solution, it comprises all solutions derived from differential equations, representing an entire family of functions. It is constituted by incorporating arbitrary constants which tailor the function for specific initial conditions or parameters.
General solutions are crucial because they describe the general behavior of the equation without specifying conditions. For example, the general solution might take the form \(y = c_1 t^2 + c_2 t^{-1} + 2t + 1\), indicating a family of possible solutions depending on constants \(c_1\) and \(c_2\). This family can represent diverse scenarios based on varying initial conditions.

Nonhomogeneous Differential Equation

A nonhomogeneous differential equation is a type where an external force or input, typically expressed as a non-zero function, influences its behavior. In our case, the Euler's equation \(t^2 y'' + \alpha t y' + \beta y = g(t)\) is nonhomogeneous since it equals a non-zero function \(g(t)\). Such equations contrast with homogeneous ones, where \(g(t)\) would be zero.
The solution to nonhomogeneous equations usually involves finding both a particular solution and a homogeneous solution, collectively making up the general solution. These components help address any effects external forces have on the system. The particular solution responds specifically to the nonhomogeneous part \(g(t)\), while the general solution considers the overall equation.

Power Rule for Differentiation

The power rule for differentiation is a fundamental concept in calculus, used to find the derivative of functions where the variable is raised to a power. Specifically, this rule states that the derivative of \(x^n\) is \(n x^{n-1}\). It plays a vital role in calculating the derivatives needed for solving differential equations. In our exercise, where the derived function \(y(t) = c_1 t^2 + c_2 t^{-1} + 2t + 1\) needs differentiation, the power rule makes this process straightforward.

• For \(c_1 t^2\), the derivative is \(2c_1 t\).
• For \(c_2 t^{-1}\), it becomes \(-c_2 t^{-2}\).
• For linear terms like \(2t\), the derivative is simply 2.

Utilizing the power rule ensures accurate determination of derivatives, serving as a backbone in manipulating equations for solutions. It is instrumental, not just for theoretical concepts but also for practical application in numerous fields.