31.6.6 problem 6

Internal problem ID [5755]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 6
Date solved : Monday, January 27, 2025 at 01:12:50 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 80

dsolve(x=a*diff(y(x),x)+b*(diff(y(x),x))^2,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {\left (a^{2}+4 b x \right )^{{3}/{2}}+12 c_{1} b^{2}-6 b x a}{12 b^{2}} \\ y \left (x \right ) &= \frac {12 c_{1} b^{2}-a^{2} \sqrt {a^{2}+4 b x}-6 b x a -4 b x \sqrt {a^{2}+4 b x}}{12 b^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 74

DSolve[x==a*D[y[x],x]+b*(D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\left (a^2+4 b x\right )^{3/2}-6 a b x+12 b^2 c_1}{12 b^2} \\ y(x)\to -\frac {\frac {\left (a^2+4 b x\right )^{3/2}}{6 b}+a x}{2 b}+c_1 \\ \end{align*}