problem 741

Internal problem ID [10752]
Book : Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear ﬁrst order
Problem number : 741
Date solved : Tuesday, January 28, 2025 at 05:10:38 PM
CAS classiﬁcation : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {\left (a y^{2}+b \,x^{2}\right )^{3} x}{a^{{5}/{2}} \left (a y^{2}+b \,x^{2}+a \right ) y} \end{align*}

\[ \frac {\int _{\textit {\_b}}^{x}\frac {\left (b \,\textit {\_a}^{2}+a y^{2}\right )^{3} \textit {\_a}}{b \left (1+y^{2}\right ) a^{{5}/{2}}+a^{{3}/{2}} b^{2} \textit {\_a}^{2}+\left (b \,\textit {\_a}^{2}+a y^{2}\right )^{3}}d \textit {\_a}}{a^{3}}-\frac {\int _{}^{y}\frac {2 \left (\left (b \left (\textit {\_f}^{2}+1\right ) a^{{5}/{2}}+a^{{3}/{2}} b^{2} x^{2}+\left (\textit {\_f}^{2} a +b \,x^{2}\right )^{3}\right ) b \left (\int _{\textit {\_b}}^{x}\frac {\left (b \,\textit {\_a}^{2}+\textit {\_f}^{2} a \right )^{2} \textit {\_a} \left (2 b \,\textit {\_a}^{2}+2 \textit {\_f}^{2} a +3 a \right )}{{\left (b \left (\textit {\_f}^{2}+1\right ) a^{{5}/{2}}+a^{{3}/{2}} b^{2} \textit {\_a}^{2}+\left (b \,\textit {\_a}^{2}+\textit {\_f}^{2} a \right )^{3}\right )}^{2}}d \textit {\_a} \right )+\frac {b \,x^{2}}{2}+\frac {a \left (\textit {\_f}^{2}+1\right )}{2}\right ) \textit {\_f}}{b \left (\textit {\_f}^{2}+1\right ) a^{{5}/{2}}+a^{{3}/{2}} b^{2} x^{2}+\left (\textit {\_f}^{2} a +b \,x^{2}\right )^{3}}d \textit {\_f}}{\sqrt {a}}+c_{1} = 0 \]

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {-K[2]^3 a^{7/2}-K[2] a^{7/2}-b x^2 K[2] a^{5/2}}{b^3 x^6+3 a b^2 K[2]^2 x^4+3 a^2 b K[2]^4 x^2+a^{3/2} b^2 x^2+a^3 K[2]^6+a^{5/2} b K[2]^2+a^{5/2} b}-\int _1^x\left (-\frac {2 b K[1] K[2] a^{5/2}}{b^3 K[1]^6+3 a b^2 K[2]^2 K[1]^4+3 a^2 b K[2]^4 K[1]^2+a^{3/2} b^2 K[1]^2+a^3 K[2]^6+a^{5/2} b K[2]^2+a^{5/2} b}-\frac {\left (-a^{3/2} b^2 K[1]^3-a^{5/2} b K[2]^2 K[1]-a^{5/2} b K[1]\right ) \left (6 a^3 K[2]^5+12 a^2 b K[1]^2 K[2]^3+6 a b^2 K[1]^4 K[2]+2 a^{5/2} b K[2]\right )}{\left (b^3 K[1]^6+3 a b^2 K[2]^2 K[1]^4+3 a^2 b K[2]^4 K[1]^2+a^{3/2} b^2 K[1]^2+a^3 K[2]^6+a^{5/2} b K[2]^2+a^{5/2} b\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (K[1]+\frac {-a^{3/2} b^2 K[1]^3-a^{5/2} b y(x)^2 K[1]-a^{5/2} b K[1]}{b^3 K[1]^6+3 a b^2 y(x)^2 K[1]^4+3 a^2 b y(x)^4 K[1]^2+a^{3/2} b^2 K[1]^2+a^3 y(x)^6+a^{5/2} b y(x)^2+a^{5/2} b}\right )dK[1]=c_1,y(x)\right ] \]

60.2.165 problem 741