77.1.30 problem 47 (page 56)

Internal problem ID [17920]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 47 (page 56)
Date solved : Tuesday, January 28, 2025 at 08:28:17 PM
CAS classification : [_rational]

\begin{align*} y^{\prime } \left (x^{2}+y^{2}+3\right )&=2 x \left (2 y-\frac {x^{2}}{y}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 3283

dsolve(diff(y(x),x)*(x^2+y(x)^2+3)=2*x*(2*y(x)-x^2/y(x)),y(x), singsol=all)
 

\begin{align*} y &= -\frac {\sqrt {3}\, \sqrt {\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (-\sqrt {3}\, \left (\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+4 c_{1} \right ) \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}-4 c_{1}^{4}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{3}+\left (-\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}+36 x^{2}+72\right ) c_{1}^{2}+9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right ) c_{1} +3 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (2 x^{2}+3\right )\right )}}{3 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}} \\ y &= \frac {\sqrt {3}\, \sqrt {\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (-\sqrt {3}\, \left (\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+4 c_{1} \right ) \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}-4 c_{1}^{4}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{3}+\left (-\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}+36 x^{2}+72\right ) c_{1}^{2}+9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right ) c_{1} +3 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (2 x^{2}+3\right )\right )}}{3 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}} \\ y &= -\frac {\sqrt {6}\, \sqrt {\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (\left (\sqrt {3}\, \left (\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+4 c_{1} \right )-3 i \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+12 i c_{1} \right ) \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}+i c_{1} \left (4 c_{1}^{3}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{2}+\left (-36 x^{2}-72\right ) c_{1} +9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right )\right ) \sqrt {3}+4 c_{1}^{4}+2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{3}+\left (-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}-36 x^{2}-72\right ) c_{1}^{2}-9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right ) c_{1} +6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (2 x^{2}+3\right )\right )}}{6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}} \\ y &= \frac {\sqrt {6}\, \sqrt {\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (\left (\sqrt {3}\, \left (\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+4 c_{1} \right )-3 i \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+12 i c_{1} \right ) \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}+i c_{1} \left (4 c_{1}^{3}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{2}+\left (-36 x^{2}-72\right ) c_{1} +9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right )\right ) \sqrt {3}+4 c_{1}^{4}+2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{3}+\left (-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}-36 x^{2}-72\right ) c_{1}^{2}-9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right ) c_{1} +6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (2 x^{2}+3\right )\right )}}{6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}} \\ y &= -\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {-\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (\left (\left (-\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}-4 c_{1} \right ) \sqrt {3}-3 i \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+12 i c_{1} \right ) \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}+i c_{1} \left (4 c_{1}^{3}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{2}+\left (-36 x^{2}-72\right ) c_{1} +9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right )\right ) \sqrt {3}-4 c_{1}^{4}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{3}+\left (2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}+36 x^{2}+72\right ) c_{1}^{2}+9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right ) c_{1} -6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (2 x^{2}+3\right )\right )}}{6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}} \\ y &= \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {-\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (\left (\left (-\left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}-4 c_{1} \right ) \sqrt {3}-3 i \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}}+12 i c_{1} \right ) \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}+i c_{1} \left (4 c_{1}^{3}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{2}+\left (-36 x^{2}-72\right ) c_{1} +9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right )\right ) \sqrt {3}-4 c_{1}^{4}-2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} c_{1}^{3}+\left (2 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}+36 x^{2}+72\right ) c_{1}^{2}+9 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{1}/{3}} \left (x^{2}+2\right ) c_{1} -6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}} \left (2 x^{2}+3\right )\right )}}{6 \left (-108 c_{1} x^{2}-216 c_{1} +8 c_{1}^{3}+12 \sqrt {3}\, \sqrt {\left (-4 x^{2}-8\right ) c_{1}^{4}+27 \left (x^{2}+2\right )^{2} c_{1}^{2}}\right )^{{2}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.202 (sec). Leaf size: 2419

DSolve[D[y[x],x]*(x^2+y[x]^2+3)==2*x*(2*y[x]-x^2/y[x]),y[x],x,IncludeSingularSolutions -> True]
 

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