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Chapter 8: Problem 12

$$x(x+1) y^{\prime \prime}+(x+5) y^{\prime}-4 y=0$$

Short Answer

Expert verified
The general solution to the equation will be of the form \(y = c_1 x^{r_1} + c_2 x^{r_2}\) where \(r_1\) and \(r_2\) are the roots found in step 4 and \(c_1\) and \(c_2\) are constants determined by initial conditions.

Step by step solution

01

Assume a solution form

Assume that a solution to the equation is of the form \(y=x^r\). We are making this assumption based on the structure of the differential equation.
02

Derive necessary components

Find necessary components such as the first and the second derivative of the assumed solution. Here, the first derivative \(y'\) equals \(r x^{r-1}\) and the second derivative \(y''\) equals \(r(r-1) x^{r-2}\).
03

Substitute solution into the equation

Substitute \(y\), \(y'\), and \(y''\) from the assumed solution into the equation: \(x(x+1)r(r-1)x^{r-2} + (x+5)rx^{r-1} - 4x^r = 0\).
04

Simplification and finding roots

Simplify the equation, factoring out \(x^{r}\) to end up with a quadratic equation in terms of \(r\). Since the factor involving \(r\) must be zero for all \(x\), set the factor equal to zero and solve for \(r\). This will give two roots \(r_1\) and \(r_2\).
05

Write down the general solution

The general solution of the Cauchy-Euler differential equation is \(y = c_1 x^{r_1} + c_2 x^{r_2}\), where \(c_1\) and \(c_2\) are constants determined by initial conditions.

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