Problems

01

Power series solution of second-order ODE

Find the complete family of power series solutions to $y^{\prime \prime }+x{y}^{\prime }+y=0$.

02

Power series solution of second-order ODE

Find the complete family of power series solutions to $y^{\prime \prime }+y^{\prime }+x^{2}y=1$.

03

Power series solution of second-order ODE

Find the complete family of power series solutions to $(1+x^2)y^{\prime \prime }+x{y}^{\prime }+y=0$.

04

Chebyshev polynomials

Consider the Chebyshev ODE given by $(1-x^2)y^{\prime \prime }-x{y}^{\prime }+p^{2}y=0$, where $p\in \mathbb{R}$ is some constant. Find two independent power series solutions to this ODE (expanded around $x=0$), and verify that when $p=n\in \mathbb{N}$ is an integer, then these series terminate, thus giving polynomials.

05

Alternative power series method: using initial conditions

Consider the IVP given by $y^{\prime \prime }+y^{\prime }-xy=0$, $y(0)=0$, $y^{\prime }(0)=2$. Notice that the equation directly determines $y^{\prime \prime }(0)$.

Now differentiate the equation: $y^{\prime \prime \prime }+y^{\prime \prime }-y-x{y}^{\prime }=0$. Notice that this equation now determines $y^{\prime \prime \prime }(0)$ in terms of the initial conditions plus the information of $y^{\prime \prime }(0)$.

We can iterate the process to compute every $y^{(n)}(0)$. Recall that $a_n=\frac{y^{(n)}(0)}{n!}$. We therefore have a new procedure to calculate every $a_n$. Follow this procedure to compute the first 7 terms $a_0,\dots ,a_6$.

06

Power series: using series multiplication

Find the first 6 terms of a power series solution to the IVP given by $y^{\prime }+(\sin x)y=0$, $y(0)=1$. [Hint: substitute in the power series for $\sin x$.]