72.2.9 problem 9

Internal problem ID [14644]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 06:46:01 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\left (y+\frac {1}{2}\right ) \left (y+t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.368 (sec). Leaf size: 92

dsolve([diff(y(t),t)=(y(t)+1/2)*(y(t)+t),y(0) = 1/2],y(t), singsol=all)
 

\[ y = \frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {1}{8}} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}}{4}\right )+\sqrt {\pi }\, {\mathrm e}^{-\frac {1}{8}} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (2 t -1\right )}{4}\right )+4 i {\mathrm e}^{\frac {t \left (t -1\right )}{2}}-2 i}{-2 \sqrt {\pi }\, {\mathrm e}^{-\frac {1}{8}} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}}{4}\right )-2 \sqrt {\pi }\, {\mathrm e}^{-\frac {1}{8}} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (2 t -1\right )}{4}\right )+4 i} \]

✓ Solution by Mathematica

Time used: 0.378 (sec). Leaf size: 65

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

\[ y(t)\to -\frac {\int _0^te^{\frac {1}{2} (K[1]-1) K[1]}dK[1]+2 e^{\frac {1}{2} (t-1) t}-1}{2 \left (\int _0^te^{\frac {1}{2} (K[1]-1) K[1]}dK[1]-1\right )} \]