Internal problem ID [3487]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 754.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+a \,x^{2}+b y=0} \end {gather*}
✗ Solution by Maple
dsolve(diff(y(x),x)^2+a*x^2+b*y(x) = 0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 74.447 (sec). Leaf size: 2397
DSolve[(y'[x])^2+a x^2+b y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{4} \left (\frac {4 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {-a x^2-b y(x)}}{\sqrt {a} x}\right )}{\sqrt {-a}}+\frac {2 b \sqrt {b^2-16 a} \text {ArcTan}\left (\frac {b x-4 \sqrt {-a x^2-b y(x)}}{\sqrt {16 a-b^2} x}\right )}{\sqrt {-\left (b^2-16 a\right )^2}}+\frac {2 b \sqrt {b^2-16 a} \text {ArcTan}\left (\frac {b x+4 \sqrt {-a x^2-b y(x)}}{\sqrt {16 a-b^2} x}\right )}{\sqrt {-\left (b^2-16 a\right )^2}}+4 \tanh ^{-1}\left (\frac {a \sqrt {-a x^2-b y(x)}}{(-a)^{3/2} x}\right )-2 \tanh ^{-1}\left (\frac {x \sqrt {-a x^2-b y(x)}}{2 y(x)}\right )-\frac {2 b \sqrt {-\left (b^2-16 a\right )^2} \tanh ^{-1}\left (\frac {b x-4 \sqrt {-a x^2-b y(x)}}{\sqrt {b^2-16 a} x}\right )}{\left (16 a-b^2\right )^{3/2}}-\frac {2 b \sqrt {-\left (b^2-16 a\right )^2} \tanh ^{-1}\left (\frac {b x+4 \sqrt {-a x^2-b y(x)}}{\sqrt {b^2-16 a} x}\right )}{\left (16 a-b^2\right )^{3/2}}+2 \log (y(x))-2 \log (-b y(x))-\log \left (b \left (x \sqrt {-a x^2-b y(x)}-2 y(x)\right )\right )+\log \left (-b \left (\sqrt {-a x^2-b y(x)} x+2 y(x)\right )\right )\right )+\frac {1}{4} \left (\frac {2 \sqrt {-a} \left (4 \sqrt {a}-\sqrt {16 a-b^2}\right ) \sqrt {-b^2+32 a+8 \sqrt {a} \sqrt {16 a-b^2}} \text {ArcTan}\left (\frac {-2 a x^2-2 \sqrt {-a} \sqrt {-a x^2-b y(x)} x-b y(x)}{\sqrt {-b^2+32 a+8 \sqrt {a} \sqrt {16 a-b^2}} y(x)}\right )}{\sqrt {a} b^2}+\frac {2 \sqrt {-a} \left (4 \sqrt {a}+\sqrt {16 a-b^2}\right ) \sqrt {b^2-32 a+8 \sqrt {a} \sqrt {16 a-b^2}} \tanh ^{-1}\left (\frac {-2 a x^2-2 \sqrt {-a} \sqrt {-a x^2-b y(x)} x-b y(x)}{\sqrt {b^2-32 a+8 \sqrt {a} \sqrt {16 a-b^2}} y(x)}\right )}{\sqrt {a} b^2}+\frac {2 \sqrt {-a} b \tanh ^{-1}\left (\frac {2 a^2 x^4+\sqrt {-a} b y(x) \sqrt {-a x^2-b y(x)} x+2 a \left (\sqrt {-a} \sqrt {-a x^2-b y(x)} x^3+b y(x) x^2+4 y(x)^2\right )}{2 \sqrt {a} \sqrt {16 a-b^2} y(x)^2}\right )}{\sqrt {a} \sqrt {16 a-b^2}}-4 \log \left (\sqrt {-a x^2-b y(x)}-\sqrt {-a} x\right )+\text {RootSum}\left [\text {$\#$1}^8-2 b^2 y(x)^2 \text {$\#$1}^4+64 a y(x)^2 \text {$\#$1}^4+b^4 y(x)^4\&,\frac {\log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) \text {$\#$1}^6-b \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x) \text {$\#$1}^4-b^2 \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x)^2 \text {$\#$1}^2+32 a \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x)^2 \text {$\#$1}^2+b^3 \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x)^3}{\text {$\#$1}^6-b^2 y(x)^2 \text {$\#$1}^2+32 a y(x)^2 \text {$\#$1}^2}\&\right ]\right )=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {1}{4} \left (\frac {4 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {-a x^2-b y(x)}}{\sqrt {a} x}\right )}{\sqrt {-a}}+\frac {2 b \sqrt {b^2-16 a} \text {ArcTan}\left (\frac {b x-4 \sqrt {-a x^2-b y(x)}}{\sqrt {16 a-b^2} x}\right )}{\sqrt {-\left (b^2-16 a\right )^2}}+\frac {2 b \sqrt {b^2-16 a} \text {ArcTan}\left (\frac {b x+4 \sqrt {-a x^2-b y(x)}}{\sqrt {16 a-b^2} x}\right )}{\sqrt {-\left (b^2-16 a\right )^2}}+4 \tanh ^{-1}\left (\frac {a \sqrt {-a x^2-b y(x)}}{(-a)^{3/2} x}\right )+2 \tanh ^{-1}\left (\frac {x \sqrt {-a x^2-b y(x)}}{2 y(x)}\right )-\frac {2 b \sqrt {-\left (b^2-16 a\right )^2} \tanh ^{-1}\left (\frac {b x-4 \sqrt {-a x^2-b y(x)}}{\sqrt {b^2-16 a} x}\right )}{\left (16 a-b^2\right )^{3/2}}-\frac {2 b \sqrt {-\left (b^2-16 a\right )^2} \tanh ^{-1}\left (\frac {b x+4 \sqrt {-a x^2-b y(x)}}{\sqrt {b^2-16 a} x}\right )}{\left (16 a-b^2\right )^{3/2}}+2 \log (y(x))-2 \log (-b y(x))+\log \left (b \left (x \sqrt {-a x^2-b y(x)}-2 y(x)\right )\right )-\log \left (-b \left (\sqrt {-a x^2-b y(x)} x+2 y(x)\right )\right )\right )+\frac {1}{4} \left (\frac {2 \left (\sqrt {16 a-b^2}-4 \sqrt {a}\right ) \sqrt {-b^2+32 a+8 \sqrt {a} \sqrt {16 a-b^2}} \text {ArcTan}\left (\frac {-2 a x^2-2 \sqrt {-a} \sqrt {-a x^2-b y(x)} x-b y(x)}{\sqrt {-b^2+32 a+8 \sqrt {a} \sqrt {16 a-b^2}} y(x)}\right ) a^{3/2}}{(-a)^{3/2} b^2}+\frac {2 \left (4 \sqrt {a}+\sqrt {16 a-b^2}\right ) \sqrt {b^2-32 a+8 \sqrt {a} \sqrt {16 a-b^2}} \tanh ^{-1}\left (\frac {-2 a x^2-2 \sqrt {-a} \sqrt {-a x^2-b y(x)} x-b y(x)}{\sqrt {b^2-32 a+8 \sqrt {a} \sqrt {16 a-b^2}} y(x)}\right ) \sqrt {a}}{\sqrt {-a} b^2}+\frac {2 b \tanh ^{-1}\left (\frac {2 a^2 x^4+\sqrt {-a} b y(x) \sqrt {-a x^2-b y(x)} x+2 a \left (\sqrt {-a} \sqrt {-a x^2-b y(x)} x^3+b y(x) x^2+4 y(x)^2\right )}{2 \sqrt {a} \sqrt {16 a-b^2} y(x)^2}\right ) \sqrt {a}}{\sqrt {-a} \sqrt {16 a-b^2}}-4 \log \left (\sqrt {-a x^2-b y(x)}-\sqrt {-a} x\right )+\text {RootSum}\left [\text {$\#$1}^8-2 b^2 y(x)^2 \text {$\#$1}^4+64 a y(x)^2 \text {$\#$1}^4+b^4 y(x)^4\&,\frac {\log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) \text {$\#$1}^6-b \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x) \text {$\#$1}^4-b^2 \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x)^2 \text {$\#$1}^2+32 a \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x)^2 \text {$\#$1}^2+b^3 \log \left (-\sqrt {-a} x-\text {$\#$1}+\sqrt {-a x^2-b y(x)}\right ) y(x)^3}{\text {$\#$1}^6-b^2 y(x)^2 \text {$\#$1}^2+32 a y(x)^2 \text {$\#$1}^2}\&\right ]\right )=c_1,y(x)\right ] \\ \end{align*}