74.3.6 problem 6

Internal problem ID [15866]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 08:10:42 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=4 t^{2}-t y^{2} \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.301 (sec). Leaf size: 127

dsolve([diff(y(t),t)=4*t^2-t*y(t)^2,y(2) = 1],y(t), singsol=all)
 

\[ y = -\frac {2 \sqrt {t}\, \left (\left (2 \sqrt {2}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right )+\operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (-\frac {3}{5}, \frac {4 t^{{5}/{2}}}{5}\right )+\operatorname {BesselK}\left (\frac {3}{5}, \frac {4 t^{{5}/{2}}}{5}\right ) \left (-2 \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}+\operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right )\right )}{\left (-2 \sqrt {2}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right )-\operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (\frac {2}{5}, \frac {4 t^{{5}/{2}}}{5}\right )+\operatorname {BesselK}\left (\frac {2}{5}, \frac {4 t^{{5}/{2}}}{5}\right ) \left (-2 \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}+\operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right )} \]

✓ Solution by Mathematica

Time used: 0.256 (sec). Leaf size: 709

DSolve[{D[y[t],t]==4*t^2-t*y[t]^2,{y[2]==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {-3 t^{5/2} \operatorname {BesselI}\left (\frac {2}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {3}{5},\frac {4 t^{5/2}}{5}\right )+4 \sqrt {2} t^{5/2} \operatorname {BesselI}\left (-\frac {3}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {3}{5},\frac {4 t^{5/2}}{5}\right )-4 \sqrt {2} t^{5/2} \operatorname {BesselI}\left (\frac {3}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {7}{5},\frac {4 t^{5/2}}{5}\right )+3 t^{5/2} \operatorname {BesselI}\left (-\frac {2}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {7}{5},\frac {4 t^{5/2}}{5}\right )-4 \sqrt {2} t^{5/2} \operatorname {BesselI}\left (-\frac {7}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {7}{5},\frac {4 t^{5/2}}{5}\right )+t^{5/2} \left (4 \sqrt {2} \operatorname {BesselI}\left (-\frac {3}{5},\frac {16 \sqrt {2}}{5}\right )-3 \operatorname {BesselI}\left (\frac {2}{5},\frac {16 \sqrt {2}}{5}\right )+4 \sqrt {2} \operatorname {BesselI}\left (\frac {7}{5},\frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (-\frac {7}{5},\frac {4 t^{5/2}}{5}\right )+4 \sqrt {2} t^{5/2} \operatorname {BesselI}\left (\frac {7}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {3}{5},\frac {4 t^{5/2}}{5}\right )+t^{5/2} \left (-4 \sqrt {2} \operatorname {BesselI}\left (-\frac {7}{5},\frac {16 \sqrt {2}}{5}\right )+3 \operatorname {BesselI}\left (-\frac {2}{5},\frac {16 \sqrt {2}}{5}\right )-4 \sqrt {2} \operatorname {BesselI}\left (\frac {3}{5},\frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (-\frac {3}{5},\frac {4 t^{5/2}}{5}\right )+4 \sqrt {2} \operatorname {BesselI}\left (-\frac {3}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {4 t^{5/2}}{5}\right )+3 \operatorname {BesselI}\left (-\frac {2}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {4 t^{5/2}}{5}\right )-4 \sqrt {2} \operatorname {BesselI}\left (-\frac {7}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {4 t^{5/2}}{5}\right )+4 \sqrt {2} \operatorname {BesselI}\left (\frac {7}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {4 t^{5/2}}{5}\right )-4 \sqrt {2} \operatorname {BesselI}\left (\frac {3}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {4 t^{5/2}}{5}\right )-3 \operatorname {BesselI}\left (\frac {2}{5},\frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {4 t^{5/2}}{5}\right )}{t^2 \left (\left (4 \sqrt {2} \operatorname {BesselI}\left (-\frac {3}{5},\frac {16 \sqrt {2}}{5}\right )-3 \operatorname {BesselI}\left (\frac {2}{5},\frac {16 \sqrt {2}}{5}\right )+4 \sqrt {2} \operatorname {BesselI}\left (\frac {7}{5},\frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {4 t^{5/2}}{5}\right )+\left (-4 \sqrt {2} \operatorname {BesselI}\left (-\frac {7}{5},\frac {16 \sqrt {2}}{5}\right )+3 \operatorname {BesselI}\left (-\frac {2}{5},\frac {16 \sqrt {2}}{5}\right )-4 \sqrt {2} \operatorname {BesselI}\left (\frac {3}{5},\frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {4 t^{5/2}}{5}\right )\right )} \]