Question

Consider the differential equation \(y^{\prime \prime}+a y^{\prime}+9 y=0\), where \(a\) is a real constant. Suppose we know that the Wronskian of a fundamental set of solutions for this equation is constant. What is the general solution for this equation?

Short answer

Answer: The general solution depends on the type of roots of the characteristic equation. 1. If \(\Delta > 0\), the general solution is \(y(x) = c_1e^{m_1x} + c_2e^{m_2x}\), where \(m_1\) and \(m_2\) are the distinct real roots. 2. If \(\Delta = 0\), the general solution is \(y(x) = c_1e^{mx} + c_2xe^{mx}\), where \(m\) is the equal real root. 3. If \(\Delta < 0\), the general solution is \(y(x) = e^{\alpha x} (c_1 \cos(\beta x) + c_2 \sin(\beta x))\), where \(m = \alpha \pm \beta i\) are the complex roots. In each case, the Wronskian of a fundamental set of solutions is constant.

Step-by-step solution

Write down the given differential equation

The given differential equation is \(y^{\prime \prime}+ay^{\prime}+9y=0\) where \(a\) is a real constant.

Find the characteristic equation

To find the characteristic equation, we replace \(y^{\prime \prime}\) with \(m^2\), \(y^{\prime}\) with \(m\), and \(y\) with 1. So, the characteristic equation becomes: \(m^2 + am + 9 = 0\).

Determine the type of roots

The type of roots of the characteristic equation depends on the discriminant \(\Delta\), where \(\Delta = a^2 - 4(1)(9) = a^2 - 36\). We will consider the following cases: 1. \(\Delta > 0\): We get two distinct real roots. 2. \(\Delta = 0\): We get two equal real roots. 3. \(\Delta < 0\): We get two complex roots.

Find the general solution depending on the type of roots

We will now find the general solution for each case mentioned above: 1. \(\Delta > 0\): The general solution is given by \(y(x) = c_1e^{m_1x} + c_2e^{m_2x}\), where \(m_1\) and \(m_2\) are the distinct real roots of the characteristic equation. 2. \(\Delta = 0\): The general solution is given by \(y(x) = c_1e^{mx} + c_2xe^{mx}\), where \(m\) is the equal real root of the characteristic equation. 3. \(\Delta < 0\): The general solution is given by \(y(x) = e^{\alpha x} (c_1 \cos(\beta x) + c_2 \sin(\beta x))\), where \(m = \alpha \pm \beta i\) are the complex roots of the characteristic equation.

Verify Wronskian is constant

Given that the Wronskian of a fundamental set of solutions is constant, which is true for linear differential equations with constant coefficients, the obtained solutions in each of the cases above satisfy this condition. So, the general solution for the equation \(y^{\prime \prime}+ay^{\prime}+9y=0\) is found for each case mentioned in Step 4.

Key concepts

Characteristic Equation

The characteristic equation is a pivotal concept in solving linear differential equations with constant coefficients. In such equations, the solution approach often involves converting the differential equation into an algebraic equation that can be solved for roots, which then inform the general solution. For differential equations like the one provided, which is of the form \(y^{\' \' }+ay^{\' }+9y=0\), we rewrite the equation in terms of a new variable, typically \(m\), representing the derivative of \(y\). Thus, the characteristic equation for this problem is \(m^2 + am + 9 = 0\).

Wronskian

The Wronskian is an essential tool in the analysis of differential equations. It is a determinant that helps in determining if a set of functions are linearly independent, which is a crucial criterion for their eligibility as solutions to the differential equation. Specifically, for a pair of solutions to a second-order differential equation, the Wronskian will be constant if and only if these solutions are linearly independent and thereby form a fundamental set of solutions. In the given differential equation, the Wronskian being constant indicates that any two solutions we find will form a valid fundamental set, translating to a correctly formed general solution.

Complex Roots

When the characteristic equation yields complex roots, the general solution to the differential equation takes a form that reflects the oscillatory nature of complex numbers. For the characteristic equation \(m^2 + am + 9 = 0\), if the discriminant \(\Delta = a^2 - 36\) is less than zero, it means we have complex roots, which can be represented as \(\alpha \pm \beta i\). This leads to a general solution of the differential equation which involves combinations of exponential, sine, and cosine functions, represented as \(y(x) = e^{\alpha x} (c_1 \cos(\beta x) + c_2 \sin(\beta x))\), where \(c_1\) and \(c_2\) are constants determined by initial conditions or boundary values.

Discriminant

The discriminant, denoted as \(\Delta\), is a critical value calculated from the coefficients of a quadratic characteristic equation. It dictates the nature of the roots, and thus, influences the form of the general solution to the differential equation. The discriminant is given by \(\Delta = a^2 - 4ac\), where \(a\) and \(c\) are coefficients from the characteristic equation \(m^2 + am + 9 = 0\). The sign of the discriminant reveals whether we have two distinct real roots (\(\Delta > 0\)), a repeated real root (\(\Delta = 0\)), or two complex roots (\(\Delta < 0\)). This understanding of the discriminant empowers us to structure the general solution accordingly, ensuring it represents the behavior of the differential equation's solutions accurately.