73.8.44 problem 13.7 (d)

Internal problem ID [15252]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.7 (d)
Date solved : Tuesday, January 28, 2025 at 07:50:56 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }+2 {y^{\prime }}^{2}&=3 y^{\prime } y \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&={\frac {3}{4}} \end{align*}

✓ Solution by Maple

Time used: 0.223 (sec). Leaf size: 14

dsolve([y(x)*diff(y(x),x$2)+2*diff(y(x),x)^2=3*y(x)*diff(y(x),x),y(0) = 2, D(y)(0) = 3/4],y(x), singsol=all)
 

\[ y = \left (3 \,{\mathrm e}^{3 x}+5\right )^{{1}/{3}} \]

✓ Solution by Mathematica

Time used: 0.495 (sec). Leaf size: 118

DSolve[{y[x]*D[y[x],{x,2}]+2*D[y[x],x]^2==3*y[x]*D[y[x],x],{y[0]==2,Derivative[1][y][0] ==3/4}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ]\left [\int _1^{\frac {3}{8}}\frac {1}{(K[1]-1) K[1]}dK[1]-3 K[2]\right ]dK[2]-\int _1^0\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ]\left [\int _1^{\frac {3}{8}}\frac {1}{(K[1]-1) K[1]}dK[1]-3 K[2]\right ]dK[2]\right ) \]