36.4.6 problem 7

Internal problem ID [6344]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problems. page 79
Problem number : 7
Date solved : Monday, January 27, 2025 at 01:57:43 PM
CAS classification : [_separable]

\begin{align*} t^{3} y^{2}+\frac {t^{4} y^{\prime }}{y^{6}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 105

dsolve(t^3*y(t)^2+t^4/(y(t)^6)*diff(y(t),t)=0,y(t), singsol=all)
 

\begin{align*} y &= \frac {1}{\left (c_1 +7 \ln \left (t \right )\right )^{{1}/{7}}} \\ y &= -\frac {\left (-1\right )^{{1}/{7}}}{\left (c_1 +7 \ln \left (t \right )\right )^{{1}/{7}}} \\ y &= \frac {\left (-1\right )^{{6}/{7}}}{\left (c_1 +7 \ln \left (t \right )\right )^{{1}/{7}}} \\ y &= -\frac {\left (-1\right )^{{5}/{7}}}{\left (c_1 +7 \ln \left (t \right )\right )^{{1}/{7}}} \\ y &= \frac {\left (-1\right )^{{2}/{7}}}{\left (c_1 +7 \ln \left (t \right )\right )^{{1}/{7}}} \\ y &= -\frac {\left (-1\right )^{{3}/{7}}}{\left (c_1 +7 \ln \left (t \right )\right )^{{1}/{7}}} \\ y &= \frac {\left (-1\right )^{{4}/{7}}}{\left (c_1 +7 \ln \left (t \right )\right )^{{1}/{7}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.185 (sec). Leaf size: 183

DSolve[t^3*y[t]^2+t^4/(y[t]^6)*D[y[t],t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to -\frac {\sqrt [7]{-\frac {1}{7}}}{\sqrt [7]{\log (t)-c_1}} \\ y(t)\to \frac {1}{\sqrt [7]{7} \sqrt [7]{\log (t)-c_1}} \\ y(t)\to \frac {(-1)^{2/7}}{\sqrt [7]{7} \sqrt [7]{\log (t)-c_1}} \\ y(t)\to -\frac {(-1)^{3/7}}{\sqrt [7]{7} \sqrt [7]{\log (t)-c_1}} \\ y(t)\to \frac {(-1)^{4/7}}{\sqrt [7]{7} \sqrt [7]{\log (t)-c_1}} \\ y(t)\to -\frac {(-1)^{5/7}}{\sqrt [7]{7} \sqrt [7]{\log (t)-c_1}} \\ y(t)\to \frac {(-1)^{6/7}}{\sqrt [7]{7} \sqrt [7]{\log (t)-c_1}} \\ y(t)\to 0 \\ \end{align*}