72.7.20 problem 20

Internal problem ID [14761]
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 1.9 page 133
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 07:14:26 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }&=t^{r} y+4 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 202

dsolve(diff(y(t),t)=t^r*y(t)+4,y(t), singsol=all)
 

\[ y = \frac {4 \left (\left (\frac {t \,t^{r}}{r +1}\right )^{\frac {-r -2}{2 r +2}} t^{-r} \left (r +1\right ) \left (r +2\right )^{2} \operatorname {WhittakerM}\left (\frac {r +2}{2 r +2}, \frac {2 r +3}{2 r +2}, \frac {t \,t^{r}}{r +1}\right )+\left (\left (r +2\right ) t^{-r}+t \right ) \left (r +1\right )^{2} \left (\frac {t \,t^{r}}{r +1}\right )^{\frac {-r -2}{2 r +2}} \operatorname {WhittakerM}\left (-\frac {r}{2 r +2}, \frac {2 r +3}{2 r +2}, \frac {t \,t^{r}}{r +1}\right )+\frac {c_{1} \left (r +2\right ) \left (r +\frac {3}{2}\right ) {\mathrm e}^{\frac {t^{r} t}{2 r +2}}}{2}\right ) {\mathrm e}^{\frac {t^{r} t}{2 r +2}}}{2 r^{2}+7 r +6} \]

✓ Solution by Mathematica

Time used: 0.065 (sec). Leaf size: 66

DSolve[D[y[t],t]==t^r*y[t]+4,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{\frac {t^{r+1}}{r+1}} \left (-\frac {4 t \left (\frac {t^{r+1}}{r+1}\right )^{-\frac {1}{r+1}} \Gamma \left (\frac {1}{r+1},\frac {t^{r+1}}{r+1}\right )}{r+1}+c_1\right ) \]