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Chapter 12: Problem 12

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

Short Answer

Expert verified
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

01

Identify Differential Equation Type

Determine the type of differential equation. The given equation is a linear second-order differential equation of the form: \[3xy'' - 2(3x-1)y' + (3x-2)y = 0\]
02

Standard Form of Differential Equation

Rewrite the differential equation in its standard form: \[y'' - \frac{2(3x-1)}{3x}y' + \frac{3x-2}{3x} y = 0\]
03

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: \[3x[r(r-1)x^{r-2}] - 2(3x-1)[rx^{r-1}] + (3x-2)x^r = 0\]
04

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 \(r\):
05

Compute Roots for \(r\)

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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:

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Most popular questions from this chapter

For each of the following equations, one solution \(u\) is given. Find the other solution by assuming a solution of the form \(y=u v\). \(3 x y^{\prime \prime}-2(3 x-1) y^{\prime}+(3 x-2) y=0 ; u=e^{x}\)

Find the general solutions of the following differential equations by looking them up in Kamke. \(x^{2} y^{\prime \prime}+(x+1) y^{\prime}-y=0\)

For each of the following equations, one solution \(u\) is given. Find the other solution by assuming a solution of the form \(y=u v\). \(\left(x^{2}+1\right) y^{\prime \prime}-2 x y^{\prime}+2 y=0 ; u=x\)

\(\lim _{x \rightarrow 0} J_{1}(x) / x=\frac{1}{2}\)

Solve the following differential equations by the method of Frobenius (generalized power series). \(x^{2} y^{\prime \prime}+x y^{\prime}-9 y=0\)
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