Question

(a) Find the power series in \(x\) for the general solution of \(y^{\prime \prime}+x y^{\prime}+2 y=0\). (b) For several choices of \(a_{0}\) and \(a_{1},\) use differential equations software to solve the initial value problem $$ y^{\prime \prime}+x y^{\prime}+2 y=0, \quad y(0)=a_{0}, \quad y^{\prime}(0)=a_{1} $$ numerically on (-5,5) (c) For fixed \(r\) in \\{1,2,3,4,5\\} graph $$ T_{N}(x)=\sum_{n=0}^{N} a_{n} x^{n} $$ and the solution obtained in (a) on \((-r, r)\). Continue increasing \(N\) until there's no perceptible difference between the two graphs.

Short answer

Question: Solve the differential equation \(y''+xy'+2y=0\) using the power series method and compare the solution with the numerical one for different initial conditions and on various intervals. Answer: The general solution of the given differential equation can be represented as a power series \(y(x)=\sum_{n=0}^\infty a_nx^n\), where the coefficients \(a_n\) are determined by the recurrence relation: $$a_{n+2} = -\frac{(n+1)x}{(n+2)(n+1)}a_{n+1}-\frac{2}{(n+2)(n+1)}a_n$$ For specific initial conditions and intervals, numerical solutions can be obtained using software like MATLAB or Mathematica. By computing the truncated power series and comparing the graphs of the power series solution and the numerical solution with increasing values of N, we can verify the accuracy of our power series representation.

Step-by-step solution

Write the differential equation and power series

To solve the given differential equation, let's first write it down: $$y''+xy'+2y=0.$$ Now, we will find a power series representation of y(x): $$y(x) = \sum_{n=0}^{\infty} a_nx^n$$

Calculate derivatives

Next, we will find the first and second derivatives of y(x) and substitute them back into the differential equation: $$y'(x) = \sum_{n=1}^{\infty} na_nx^{n-1} \quad \text{and} \quad y''(x) = \sum_{n=2}^{\infty} n(n-1)a_nx^{n-2}$$

Substitute derivatives into the differential equation

Now, we substitute the derivatives of y(x) in the given differential equation: $$\sum_{n=2}^{\infty} n(n-1)a_nx^{n-2} + x\sum_{n=1}^{\infty} na_nx^{n-1} + 2\sum_{n=0}^{\infty} a_nx^n = 0$$

Change the index of summation for proper alignment

We need the indices of summation to be the same in all three terms of the equation. Thus, make the substitution \(n \rightarrow n+2\): $$\sum_{n=0}^{\infty} (n+2)(n+1)a_{n+2}x^n + x\sum_{n=0}^{\infty} (n+1)a_{n+1}x^n + 2\sum_{n=0}^{\infty} a_nx^n = 0$$

Find recurrence relation

Now, we can combine the summation terms to find a recurrence relation: $$\sum_{n=0}^{\infty} [(n+2)(n+1)a_{n+2}+x(n+1)a_{n+1}+2a_n]x^n = 0$$ To satisfy the above equality, all coefficients of powers of x should equal zero: $$(n+2)(n+1)a_{n+2} + (n+1)a_{n+1}x + 2a_n = 0$$ Solving for \(a_{n+2}\) gives us the recurrence relation: $$a_{n+2} = -\frac{(n+1)x}{(n+2)(n+1)}a_{n+1}-\frac{2}{(n+2)(n+1)}a_n$$

General solution of the differential equation

Using the recurrence relation, we can write the solution of the differential equation as a power series: $$y(x)=\sum_{n=0}^\infty a_nx^n$$ where the coefficients \(a_n\) are determined by the recurrence relation found above.

Compare numerical and power series solutions graphically

(b) For specific initial conditions \(a_0\) and \(a_1\), we can use software like MATLAB or Mathematica to solve the initial value problem numerically on the interval \((-5,5)\). We can also compute the first few terms of the power series solution using the recurrence relation. (c) For fixed values of \(r \in {1,2,3,4,5}\), we graph the numerical solution obtained in part (b) as well as the solution from part (a) as a truncated power series \(T_N(x) = \sum_{n=0}^N a_nx^n\), where N is sufficiently large. We increase the value of N until the difference between the graphs is negligible. By comparing the graphs, we can verify the accuracy of the power series solution compared to the numerical solution.