Internal problem ID [13521]
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: 13.8 (iv).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
\[ \boxed {y^{\prime \prime }+2 x {y^{\prime }}^{2}=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = -{\frac {1}{4}}, y^{\prime }\left (1\right ) = 5\right ] \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 29
dsolve([diff(y(x),x$2)=-2*x*diff(y(x),x)^2,y(1) = -1/4, D(y)(1) = 5],y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {5}\, x}{2}\right )}{2}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {5}}{2}\right )}{2}-\frac {1}{4} \]
✓ Solution by Mathematica
Time used: 0.185 (sec). Leaf size: 46
DSolve[{y''[x]==-2*x*y'[x]^2,{y[1]==-1/4,y'[1]==5}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} \left (-2 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{2}\right )+2 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5}}{2}\right )-1\right ) \]