60.2.38 problem 614

Internal problem ID [10625]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 614
Date solved : Tuesday, January 28, 2025 at 04:56:21 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {x \left (a -1\right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )} \end{align*}

✓ Solution by Maple

Time used: 0.090 (sec). Leaf size: 93

dsolve(diff(y(x),x) = x*(a-1)*(a+1)/(y(x)+F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)*a^2-F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)),y(x), singsol=all)
 

\begin{align*} y &= \operatorname {RootOf}\left (F \left (\frac {1}{2} \textit {\_Z}^{2}-\frac {1}{2} a^{2} x^{2}+\frac {1}{2} x^{2}\right )\right ) \\ \frac {\int _{}^{-a^{2} x^{2}+x^{2}+y^{2}}\frac {1}{F \left (\frac {\textit {\_a}}{2}\right )}d \textit {\_a} +\left (2 a^{2}-2\right ) y-2 c_{1} a^{4}+4 c_{1} a^{2}-2 c_{1}}{2 a^{4}-4 a^{2}+2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.310 (sec). Leaf size: 177

DSolve[D[y[x],x] == ((-1 + a)*(1 + a)*x)/(-F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + a^2*F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{(a-1) (a+1) F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {K[2]^2}{2}\right )}-\int _1^x\frac {K[1] K[2] F''\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {K[1]}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {y(x)^2}{2}\right )}dK[1]=c_1,y(x)\right ] \]