74.13.39 problem 67

Internal problem ID [16405]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number : 67
Date solved : Tuesday, January 28, 2025 at 09:07:33 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 26

dsolve(2*y(t)*diff(y(t),t$2)+y(t)^2=diff(y(t),t)^2,y(t), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \sqrt {c_{1}^{2}+c_{2}^{2}}+c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.607 (sec). Leaf size: 44

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

\[ y(t)\to c_2 \exp \left (\int _1^t\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [c_1-\frac {K[2]}{2}\right ]dK[2]\right ) \]