Question

\(3 x y^{\prime \prime}-2(3 x-1) y^{\prime}+(3 x-2) y=0\)

Short answer

Determine the type and standard form. Use the substitution \(y = x^r\), and simplify to find the indicial equation. Solve for \(r\).

Step-by-step solution

Identify Differential Equation Type

Determine the type of differential equation. The given equation is a linear second-order differential equation of the form: \[3xy'' - 2(3x-1)y' + (3x-2)y = 0\]

Standard Form of Differential Equation

Rewrite the differential equation in its standard form: \[y'' - \frac{2(3x-1)}{3x}y' + \frac{3x-2}{3x} y = 0\]

Find Indicial Equation

Assume a solution of the form \(y = x^r\). Substitute \(y = x^r\), \(y' = rx^{r-1}\), and \(y'' = r(r-1)x^{r-2}\) into the differential equation: \[3x[r(r-1)x^{r-2}] - 2(3x-1)[rx^{r-1}] + (3x-2)x^r = 0\]

Simplify and Solve for \(r\)

Simplify the expression to find the indicial equation - find the coefficient of the lowest power of \(x\): \[3r(r-1)x^r - 2r(3x - 1)x^r + (3x - 2)x^r= 0\] Simplifying: \[3r(r-1) - 2r(3x-1) + (3x-2) = 0\] Combine like terms and solve for \(r\):

Compute Roots for \(r\)

Solve the resulting indicial equation to find the roots \(r\).

Key concepts

standard form of differential equations

To analyze any differential equation, it is crucial to rewrite it in its standard form. This helps recognize and solve it effectively. For a linear second-order differential equation, the standard form is expressed as: