Question

Prove that the general solution of $$ (x-2) \frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}+\frac{4 y}{x^{2}}=0 $$ is given by $$ y(x)=\frac{1}{(x-2)^{2}}\left[k\left(\frac{2}{3 x}-\frac{1}{2}\right)+c x^{2}\right]. $$

Short answer

The general solution is \[ y(x) = \frac{1}{(x-2)^{2}} \left[k \left(\frac{2}{3x} - \frac{1}{2}\right) + c x^{2}\right]. \]

Step-by-step solution

Analyze the Differential Equation

Observe that the given differential equation is \[ (x-2) \frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} + \frac{4 y}{x^{2}} = 0. \] Notice that the coefficients involve variable terms like \(x-2\) and \(x^{2}\).

Propose a Substitution

To simplify the equation, use the substitution \[ y(x) = (x-2)^m \, u(x), \] where \(m\) is a constant to be determined. Differentiating, we find \[ \frac{d y}{d x} = m (x-2)^{m-1} \, u(x) + (x-2)^m \, \frac{d u}{d x}. \]

Calculate the Second Derivative

To find the second derivative, apply the product rule again: \[ \frac{d^{2} y}{d x^{2}} = m(m-1)(x-2)^{m-2} \, u(x) + 2m (x-2)^{m-1} \, \frac{d u}{d x} + (x-2)^m \, \frac{d^{2} u}{d x^{2}}. \]

Substitute Derivatives Back into the DE

Substitute the expressions for \(\frac{d y}{d x}\) and \(\frac{d^{2} y}{d x^{2}}\) back into the original differential equation and simplify: \[ (x-2) \left( m(m-1)(x-2)^{m-2} \, u(x) + 2m (x-2)^{m-1} \, \frac{d u}{d x} + (x-2)^m \, \frac{d^{2} u}{d x^{2}} \right) + 3 \left( m (x-2)^{m-1} \, u(x) + (x-2)^m \frac{d u}{d x} \right) + \frac{4 (x-2)^m \, u(x)}{x^{2}} = 0. \]

Simplify and Find the Characteristic Equation

Combine like terms and solve the characteristic equation for \(m\): \[ m(m-1) u(x) + 3m u(x) + \frac{4 u(x)}{x^2 (x-2)} = 0. \] Find the values of \(m\) that satisfy the simplified equation.

Integrate to Find General Solution

Use the roots found for \(m\) to determine the particular form of \(u(x)\). Integrate as needed to find the general solution in terms of \(u(x)\). The final solution should be of the form: \[ y(x) = \frac{1}{(x-2)^{2}} \left[k \left(\frac{2}{3x} - \frac{1}{2}\right) + c x^{2}\right]. \]

Key concepts

second-order differential equation

A second-order differential equation involves the second derivative of a function. In our case, the given differential equation is \[(x-2) \frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} + \frac{4 y}{x^{2}} = 0.\] Here, the equation includes both the first derivative \(\frac{d y}{d x}\) and the second derivative \(\frac{d^{2} y}{d x^{2}}\) of the unknown function \(y(x)\). Second-order differential equations can be more complex due to this second derivative, but they can also provide more information about the behavior of the function. To solve these equations, many techniques can be employed, including methods specific to equations with variable coefficients.

substitution method

The substitution method helps simplify complex differential equations. For this problem, we consider the substitution \(y(x) = (x-2)^m \, u(x)\), where \(m\) is a constant. This transformation changes the original function \(y(x)\), allowing us to work with a potentially simpler function \(u(x)\). By differentiating \(y(x)\) with respect to \(x\) and then substituting these derivatives back into the original equation, we transform the problem into one involving \(u(x)\) rather than \(y(x)\). This method can often reveal hidden structure and help us find solutions more easily.

variable coefficients

The given differential equation has variable coefficients, meaning the coefficients of \( \frac{d^{2} y}{d x^{2}}\) and \(\frac{d y}{d x}\) depend on the variable \(x\). For instance, in our equation, we see terms like \(x-2\) and \(x^2\) in the coefficients:\[ (x-2) \frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} + \frac{4 y}{x^{2}} = 0. \] Such equations can be more challenging to solve because their behavior changes as \(x\) varies. Using substitutions, we often convert variable coefficients into constant coefficients under the new variables, providing a more straightforward path to a solution.

general solution

Finding the general solution of a differential equation involves identifying a form that includes all possible solutions. For our differential equation:\[ y(x) = \frac{1}{(x-2)^2} \big[k \big(\frac{2}{3x} - \frac{1}{2}\big) + c x^2\big], \] where \(k\) and \(c\) are arbitrary constants. This solution fits the original equation for any values of \(k\) and \(c\). By solving the characteristic equation for \(m\), we determine the possible forms of \(u(x)\). Each value of \(m\) leads to a different solution component, and the combination of these components forms the general solution. This ensures we capture all potential behaviors of \(y(x)\) according to the differential equation.