64.4.7 problem 7

Internal problem ID [13296]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.2 (Separable equations). Exercises page 47
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 05:16:22 AM
CAS classification : [_separable]

\begin{align*} \left (x +4\right ) \left (1+y^{2}\right )+y \left (x^{2}+3 x +2\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 114

dsolve((x+4)*(y(x)^2+1) + y(x)*(x^2+3*x+2)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {-x^{6}-6 x^{5}+c_{1} x^{4}+\left (8 c_{1} +100\right ) x^{3}+\left (24 c_{1} +345\right ) x^{2}+\left (32 c_{1} +474\right ) x +16 c_{1} +239}}{\left (x +1\right )^{3}} \\ y &= -\frac {\sqrt {-x^{6}-6 x^{5}+c_{1} x^{4}+\left (8 c_{1} +100\right ) x^{3}+\left (24 c_{1} +345\right ) x^{2}+\left (32 c_{1} +474\right ) x +16 c_{1} +239}}{\left (x +1\right )^{3}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.349 (sec). Leaf size: 181

DSolve[(x+4)*(y[x]^2+1) + y[x]*(x^2+3*x+2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-1+\exp \left (2 \left (\int _1^x-\frac {K[1]+4}{K[1]^2+3 K[1]+2}dK[1]+c_1\right )\right )} \\ y(x)\to \sqrt {-1+\exp \left (2 \left (\int _1^x-\frac {K[1]+4}{K[1]^2+3 K[1]+2}dK[1]+c_1\right )\right )} \\ y(x)\to -i \\ y(x)\to i \\ y(x)\to -\sqrt {\exp \left (2 \int _1^x-\frac {K[1]+4}{K[1]^2+3 K[1]+2}dK[1]\right )-1} \\ y(x)\to \sqrt {\exp \left (2 \int _1^x-\frac {K[1]+4}{K[1]^2+3 K[1]+2}dK[1]\right )-1} \\ \end{align*}