29.22.14 problem 622

Internal problem ID [5214]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 22
Problem number : 622
Date solved : Monday, January 27, 2025 at 10:21:28 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 44

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

\[ -\ln \left (\frac {x +y \left (x \right )}{x}\right )-3 \ln \left (\frac {y \left (x \right )-3 x}{x}\right )+3 \ln \left (\frac {y \left (x \right )}{x}\right )-\ln \left (x \right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.180 (sec). Leaf size: 747

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

\begin{align*} y(x)\to \frac {1}{4} \left (-\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}-\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {\left (-8 x+e^{c_1}\right ){}^3-72 x^2 \left (-8 x+e^{c_1}\right )}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right ) \\ y(x)\to \frac {1}{4} \left (-\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}+\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {\left (-8 x+e^{c_1}\right ){}^3-72 x^2 \left (-8 x+e^{c_1}\right )}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right ) \\ y(x)\to \frac {1}{4} \left (\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}-\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {72 x^2 \left (-8 x+e^{c_1}\right )-\left (-8 x+e^{c_1}\right ){}^3}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right ) \\ y(x)\to \frac {1}{4} \left (\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}+\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {72 x^2 \left (-8 x+e^{c_1}\right )-\left (-8 x+e^{c_1}\right ){}^3}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right ) \\ \end{align*}