44.4.4 problem 1 (d)

Internal problem ID [7017]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.1 Solution curves without a solution. Exercises 2.1 at page 44
Problem number : 1 (d)
Date solved : Monday, January 27, 2025 at 02:41:05 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=x^{2}-y^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.126 (sec). Leaf size: 57

dsolve([diff(y(x),x)=x^2-y(x)^2,y(0) = 0],y(x), singsol=all)
 

\[ y = \left \{\begin {array}{cc} 0 & x =0 \\ \frac {x \left (\sqrt {2}\, \pi \operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )-2 \operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{\sqrt {2}\, \pi \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+2 \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} & \operatorname {otherwise} \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.159 (sec). Leaf size: 81

DSolve[{D[y[x],x]==x^2-y[x]^2,{y[0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {i x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )-i x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )} \]