67.2.34 problem Problem 13

Internal problem ID [13991]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 13
Date solved : Tuesday, January 28, 2025 at 08:25:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 39

dsolve(x^2*diff(y(x),x$2)-4*x^2*diff(y(x),x)+(x^2+1)*y(x)=0,y(x), singsol=all)
 

\[ y = \sqrt {x}\, {\mathrm e}^{2 x} \left (c_{1} \operatorname {BesselI}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right )+c_{2} \operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 67

DSolve[x^2*D[y[x],{x,2}]-4*x^2*D[y[x],x]+(x^2+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{2 x} \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (\frac {i \sqrt {3}}{2},-i \sqrt {3} x\right )+c_2 \operatorname {BesselY}\left (\frac {i \sqrt {3}}{2},-i \sqrt {3} x\right )\right ) \]