Internal problem ID [3992]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 751.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`]]
\[ \boxed {{y^{\prime }}^{2}-y=x^{2}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 287
dsolve(diff(y(x),x)^2 = x^2+y(x),y(x), singsol=all)
\begin{align*} 2 \sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (4 \sqrt {x^{2}+y \left (x \right )}+x \right ) \sqrt {17}}{17 x}\right )+2 \sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (-x +4 \sqrt {x^{2}+y \left (x \right )}\right ) \sqrt {17}}{17 x}\right )+2 \sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (-x^{2}+8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )+17 \ln \left (x \sqrt {x^{2}+y \left (x \right )}+2 y \left (x \right )\right )-17 \ln \left (-x \sqrt {x^{2}+y \left (x \right )}+2 y \left (x \right )\right )-17 \ln \left (-x^{4}-y \left (x \right ) x^{2}+4 y \left (x \right )^{2}\right )-c_{1} = 0 2 \sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (4 \sqrt {x^{2}+y \left (x \right )}+x \right ) \sqrt {17}}{17 x}\right )+2 \sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (-x +4 \sqrt {x^{2}+y \left (x \right )}\right ) \sqrt {17}}{17 x}\right )-2 \sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (-x^{2}+8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )+17 \ln \left (x \sqrt {x^{2}+y \left (x \right )}+2 y \left (x \right )\right )-17 \ln \left (-x \sqrt {x^{2}+y \left (x \right )}+2 y \left (x \right )\right )+17 \ln \left (-x^{4}-y \left (x \right ) x^{2}+4 y \left (x \right )^{2}\right )-c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 1.538 (sec). Leaf size: 215
DSolve[(y'[x])^2==x^2+y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{34} \left (-34 \log \left (\sqrt {x^2+y(x)}-x\right )-\left (\sqrt {17}-17\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2-\sqrt {17} y(x)+3 y(x)\right )+\left (17+\sqrt {17}\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2+\left (3+\sqrt {17}\right ) y(x)\right )\right )=c_1,y(x)\right ] \text {Solve}\left [\frac {1}{34} \left (-34 \log \left (\sqrt {x^2+y(x)}-x\right )+\left (17+\sqrt {17}\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2+\left (\sqrt {17}-5\right ) y(x)\right )-\left (\sqrt {17}-17\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2-\left (5+\sqrt {17}\right ) y(x)\right )\right )=c_1,y(x)\right ] \end{align*}