60.2.132 problem 708

Internal problem ID [10719]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 708
Date solved : Monday, January 27, 2025 at 09:30:01 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {\left (-y^{2}+4 a x \right )^{3}}{\left (-y^{2}+4 a x -1\right ) y} \end{align*}

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 747

dsolve(diff(y(x),x) = (-y(x)^2+4*a*x)^3/(-y(x)^2+4*a*x-1)/y(x),y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 0.351 (sec). Leaf size: 369

DSolve[D[y[x],x] == (4*a*x - y[x]^2)^3/(y[x]*(-1 + 4*a*x - y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {2 \left (-a K[2]^3-a K[2]+4 a^2 x K[2]\right )}{K[2]^6-12 a x K[2]^4+48 a^2 x^2 K[2]^2-2 a K[2]^2-64 a^3 x^3-2 a+8 a^2 x}-\int _1^x\left (-\frac {8 K[2] a^2}{-K[2]^6+12 a K[1] K[2]^4-48 a^2 K[1]^2 K[2]^2+2 a K[2]^2+64 a^3 K[1]^3+2 a-8 a^2 K[1]}-\frac {4 \left (4 K[1] a^3-K[2]^2 a^2-a^2\right ) \left (-6 K[2]^5+48 a K[1] K[2]^3-96 a^2 K[1]^2 K[2]+4 a K[2]\right )}{\left (-K[2]^6+12 a K[1] K[2]^4-48 a^2 K[1]^2 K[2]^2+2 a K[2]^2+64 a^3 K[1]^3+2 a-8 a^2 K[1]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (2 a+\frac {4 \left (4 K[1] a^3-y(x)^2 a^2-a^2\right )}{-y(x)^6+12 a K[1] y(x)^4-48 a^2 K[1]^2 y(x)^2+2 a y(x)^2+64 a^3 K[1]^3+2 a-8 a^2 K[1]}\right )dK[1]=c_1,y(x)\right ] \]