problem 883

Internal problem ID [10893]
Book : Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear ﬁrst order
Problem number : 883
Date solved : Tuesday, January 28, 2025 at 05:29:22 PM
CAS classiﬁcation : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+a \,x^{4} b^{2}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{{7}/{2}} y} \end{align*}

\[ \frac {\int _{\textit {\_b}}^{x}\frac {\left (b^{3} \textit {\_a}^{6}+3 a \,b^{2} \textit {\_a}^{4} y^{2}+3 a^{2} b \,\textit {\_a}^{2} y^{4}+y^{6} a^{3}+a \,b^{2} \textit {\_a}^{4}+2 a^{2} y^{2} b \,\textit {\_a}^{2}+y^{4} a^{3}+a^{3}\right ) \textit {\_a}}{y^{6} a^{3}+3 a^{2} b \,\textit {\_a}^{2} y^{4}+3 a \,b^{2} \textit {\_a}^{4} y^{2}+b^{3} \textit {\_a}^{6}+y^{4} a^{3}+2 a^{2} y^{2} b \,\textit {\_a}^{2}+a \,b^{2} \textit {\_a}^{4}+a^{3}+a^{{5}/{2}} b}d \textit {\_a}}{a^{{7}/{2}}}-\int _{}^{y}\frac {2 \textit {\_f} \left (\frac {1}{2}+\left (a^{{5}/{2}} b +b^{3} x^{6}+3 x^{4} \left (\textit {\_f}^{2}+\frac {1}{3}\right ) a \,b^{2}+3 x^{2} a^{2} \textit {\_f}^{2} \left (\textit {\_f}^{2}+\frac {2}{3}\right ) b +\left (\textit {\_f}^{6}+\textit {\_f}^{4}+1\right ) a^{3}\right ) b \left (\int _{\textit {\_b}}^{x}\frac {\left (3 b \,\textit {\_a}^{2}+3 \textit {\_f}^{2} a +2 a \right ) \left (b \,\textit {\_a}^{2}+\textit {\_f}^{2} a \right ) \textit {\_a}}{\left (a^{{5}/{2}} b +\left (\textit {\_f}^{6}+\textit {\_f}^{4}+1\right ) a^{3}+3 \textit {\_a}^{2} \left (\textit {\_f}^{2}+\frac {2}{3}\right ) \textit {\_f}^{2} b \,a^{2}+3 \textit {\_a}^{4} \left (\textit {\_f}^{2}+\frac {1}{3}\right ) b^{2} a +b^{3} \textit {\_a}^{6}\right )^{2}}d \textit {\_a} \right )\right )}{a^{{5}/{2}} b +b^{3} x^{6}+3 x^{4} \left (\textit {\_f}^{2}+\frac {1}{3}\right ) a \,b^{2}+3 x^{2} a^{2} \textit {\_f}^{2} \left (\textit {\_f}^{2}+\frac {2}{3}\right ) b +\left (\textit {\_f}^{6}+\textit {\_f}^{4}+1\right ) a^{3}}d \textit {\_f} +c_{1} = 0 \]

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

60.2.306 problem 883