Question

By making the change of variable \(x-1=t\) and assuming that \(y\) is a power series in \(t\) find two linearly independent series solutions of $$ y^{\prime \prime}+(x-1)^{2} y^{\prime}+\left(x^{2}-1\right) y=0 $$ in powers of \(x-1\). Show that you obtain the same result directly by assuming that \(y\) is a Taylor series in powers of \(x-1\) and also expressing the coefficient \(x^{2}-1\) in powers of \(x-1\).

Short answer

1. Making the change of variable and rewriting the differential equation in terms of \(t\). 2. Assuming that \(y\) is a power series in \(t\) and finding the coefficients of this series. 3. Verifying the obtained result using the method of Taylor series in powers of \(x-1\).

Step-by-step solution

Change of variable and rewriting the differential equation

Let's denote the change of variable as \(x-1=t\). This means \(x=t+1\) and \(dx=dt\). Now, let's rewrite the differential equation in terms of \(t\). $$ y^{\prime \prime}+(t+1)^{2} y^{\prime}+\left((t+1)^{2}-1\right)y=0 $$

Assume y as a power series in t and find the coefficients

Assume that \(y(t)\) is a power series in \(t\), i.e. $$ y(t) = \sum_{n=0}^\infty a_nt^n $$ Now, we have to find the derivatives \(y^{\prime}(t)\) and \(y^{\prime \prime}(t)\). $$ y^{\prime}(t) = \sum_{n=1}^\infty na_nt^{n-1} $$ $$ y^{\prime \prime}(t) = \sum_{n=2}^\infty n(n-1)a_nt^{n-2} $$ Substitute \(y(t)\), \(y^{\prime}(t)\) and \(y^{\prime \prime}(t)\) into the differential equation rewritten in terms of \(t\). $$ \sum_{n=2}^\infty n(n-1)a_nt^{n-2} + (t+1)^{2}\sum_{n=1}^\infty na_nt^{n-1} + \left((t+1)^{2}-1\right)\sum_{n=0}^\infty a_nt^n=0 $$ Now, we need to simplify the equation above and find the coefficients \(a_n\).

Verification of the result using Taylor series in powers of \(x-1\)

Express the coefficients \(x^2-1\) in powers of \(x-1\): $$ x^2 - 1 = (x-1+1)^2 - 1 = (x-1)^2 + 2(x-1) + 1 - 1 = (x-1)^2 + 2(x-1) $$ Now we can assume the \(y(t)\) to be a Taylor series in powers of \(x-1\), i.e., \(t\): $$ y(t) = \sum_{n=0}^\infty c_nt^n $$ Substitute \(y(t)\) and coefficients into the rewritten differential equation: $$ y^{\prime \prime}+(t+1)^{2}y^{\prime}+\left(t^2 + 2t\right)y=0 $$ Calculate the derivatives, substitute, and set coefficients equal to zero similarly as in Step 2.