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Magazine Chapter 8: Problem 7
Identify the singular point. Find the general solution that is valid for values of \(t\) on either side of the singular point. $$ t^{2} y^{\prime \prime}-3 t y^{\prime}+29 y=0 $$
Short Answer
Expert verified
Answer: The singular point of the given differential equation is \(t = 0\). The general solution on either side of the singular point is:
$$
y(t) = C_1 \sum_{n=0}^\infty a_n t^n + C_2 t \sum_{n=0}^\infty b_n t^n.
$$
Step by step solution
01
Check for Singular Point
First, we need to check for any singular points of the given differential equation. A singular point is a value of \(t\) where the coefficient of the highest derivative, in this case \(t^2\), becomes zero. This means we should look for values of \(t\) that satisfy the equation \(t^2 = 0\). There is only one such value: \(t = 0\).
02
Rewrite the Differential Equation
To find the general solution to the given differential equation, we need to rewrite it in a form that is easier to solve. In this case, we can rewrite it as a second-order linear homogeneous differential equation with variable coefficients by dividing the entire equation by \(t^2\):
$$
y^{\prime\prime} - \frac{3}{t}y^{\prime} + \frac{29}{t^2}y = 0.
$$
03
Use Frobenius Method to Find the General Solution
To solve this rewritten equation, we can use the Frobenius method. We make the substitution \(y(t) = t^r\sum_{n=0}^\infty a_n t^n\), where \(r\) is a constant and \(a_n\) are the coefficients we need to find. Plug this substitution into the rewritten differential equation:
$$
t^r\sum_{n=0}^\infty a_n(n + r)(n + r - 1)t^{n-1} - \frac{3}{t}t^r\sum_{n=0}^\infty a_n(n + r)t^{n-1} + \frac{29}{t^2}t^r\sum_{n=0}^\infty a_nt^n = 0.
$$
04
Simplify the Equation
Now, simplify the above equation by cancelling common terms:
$$
\sum_{n=0}^\infty a_n(n + r)(n + r - 1)t^{n+r-1} - 3\sum_{n=0}^\infty a_n(n + r)t^{n+r-2} + 29\sum_{n=0}^\infty a_nt^{n+r-2} = 0.
$$
05
Equate Coefficients and Solve for the Indicial Equation
Equate powers of \(t\) to form a recursive equation:
$$
a_0[r(r-1)] = 0 \qquad \Longrightarrow \qquad r(r-1) = 0.
$$
Solving for \(r\), we get \(r_1 = 0\) and \(r_2 = 1\).
06
Form the General Solution
Using the two values of \(r\), we can form the general solution of the differential equation as:
$$
y(t) = C_1 t^{r_1}\sum_{n=0}^\infty a_n t^n + C_2 t^{r_2}\sum_{n=0}^\infty b_n t^n.
$$
Substitute \(r_1 = 0\) and \(r_2 = 1\) into the equation to find the general solution valid for values of \(t\) on either side of the singular point \(t = 0\):
$$
y(t) = C_1 \sum_{n=0}^\infty a_n t^n + C_2 t \sum_{n=0}^\infty b_n t^n.
$$
This is the general solution to the given differential equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Singular Points
In the realm of differential equations, a singular point is a value where the differential equation ceases to be well-behaved. Specifically, for the equation we're examining, \[ t^2 y'' - 3t y' + 29y = 0, \]we identify singular points by determining where the coefficient of the highest-order derivative vanishes. Here, the highest-order derivative is associated with the term \( t^2 \).
- This makes \( t = 0 \) our singular point, as \( t^2 = 0 \) when \( t = 0 \).
Frobenius Method
The Frobenius Method is a powerful tool for finding solutions to linear differential equations when typical power series methods fail, especially near singular points. This method assumes solutions in the form of a power series, \[ y(t) = t^r \sum_{n=0}^\infty a_n t^n, \]where \( r \) is typically determined by an indicial equation derived from the differential equation.
- We substitute this series into the original differential equation.
- This substitution aids in equating coefficients of corresponding powers of \( t \).
Homogeneous Differential Equation
A homogeneous differential equation, like the one we encounter here, is a type of equation that appears without constant or external forces, typically taking the form \[ a(t) y'' + b(t) y' + c(t) y = 0. \] This implies that all terms are dependent entirely on the function \( y \) or its derivatives.
- In our example, after dividing through by \( t^2 \), creates \[ y'' - \frac{3}{t}y' + \frac{29}{t^2}y = 0. \]
- The absence of a standalone constant term ensures the equation remains homogeneous.
Indicial Equation
The indicial equation is a critical part of solving differential equations using the Frobenius method. It's derived from setting up the series substitution and ensures that the series solution is consistent near the singular point.
- In our exercise, the indicial equation emerges from equating the lowest power terms of \( t \).
- In this case, we have \[ a_0[r(r-1)] = 0. \]
- Solving this leads to \( r_1 = 0 \) and \( r_2 = 1 \), dictating the structure of our series solutions.
