60.7.162 problem 1753 (book 6.162)

Internal problem ID [11751]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1753 (book 6.162)
Date solved : Monday, January 27, 2025 at 11:34:00 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 4 y^{\prime \prime } y-5 {y^{\prime }}^{2}+a y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 6.384 (sec). Leaf size: 34

dsolve(4*diff(diff(y(x),x),x)*y(x)-5*diff(y(x),x)^2+a*y(x)^2=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \frac {16 \,{\mathrm e}^{\sqrt {a}\, x} a^{2}}{\left ({\mathrm e}^{\frac {\sqrt {a}\, x}{2}} c_{1} -c_{2} \right )^{4}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.650 (sec). Leaf size: 46

DSolve[a*y[x]^2 - 5*D[y[x],x]^2 + 4*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{a-K[1]^2}dK[1]\&\right ]\left [c_1-\frac {K[2]}{4}\right ]dK[2]\right ) \]