60.2.237 problem 813

Internal problem ID [10824]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 813
Date solved : Tuesday, January 28, 2025 at 05:18:52 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=\frac {\left (-\sqrt {a}\, x^{3}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2} \end{align*}

✓ Solution by Maple

Time used: 0.152 (sec). Leaf size: 40

dsolve(diff(y(x),x) = 1/2*(-a^(1/2)*x^3+2*(a*x^4+8*y(x))^(1/2)+2*x^2*(a*x^4+8*y(x))^(1/2)+2*x^3*(a*x^4+8*y(x))^(1/2))*a^(1/2),y(x), singsol=all)
 

\[ \frac {\sqrt {a \,x^{4}+8 y}}{4}+\frac {\left (-3 x^{4}-4 x^{3}-12 x \right ) \sqrt {a}}{12}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.915 (sec). Leaf size: 64

DSolve[D[y[x],x] == (Sqrt[a]*(-(Sqrt[a]*x^3) + 2*Sqrt[a*x^4 + 8*y[x]] + 2*x^2*Sqrt[a*x^4 + 8*y[x]] + 2*x^3*Sqrt[a*x^4 + 8*y[x]]))/2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{72} a \left (9 x^8+24 x^7+16 x^6+72 x^5+(87-72 c_1) x^4-96 c_1 x^3+144 x^2-288 c_1 x+144 c_1{}^2\right ) \]