Questions without answers in Ordinary Differential Equations

Let $W(x)$ denote the Wronskian of two linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$ of the differential equation$$(x-1) \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d...

The initial value problem$$\begin{array}{l}\dfrac{d u}{d t}=u^{2}+t^{2}, \quad t \geq 0 \\\text { with } u(0)=1\end{array}$$is solved by using the explicit Euler method w...

Let $y(x)$ be the solution of the initial value problem,$$\begin{array}{c}x^{2} y^{\prime \prime}+x y^{\prime}-y=0, \quad x>0, \\y(1)=0, \quad y^{\prime}(1)=2 .\end{array...

Let $y$ be a non-zero quadratic polynomial satisfying the differential equation\[\left(2+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-k y=0,\]where $k$ is a rea...

For some integer $k$, the differential equation\[x^{2} \frac{d^{2} y}{d x^{2}}-3 x \frac{d y}{d x}+(k+2) y=0\]is transformed into $(D-2)^{2} y=0$, where $D=\frac{d}{d t}$...

Let $y_{1}(x)$ and $y_{2}(x)$ be two linearly independent solutions of\[x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0, \quad x>0\]Let $W\left(y_{1}, y_{2}\right...