Internal problem ID [1065]
Internal file name [OUTPUT/1066_Sunday_June_05_2022_02_01_47_AM_56885570/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Exact equations. Integrating factors. Section 2.6 Page
91
Problem number: 5.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {2 y^{3}+3 y^{2} y^{\prime }=0} \]
Integrating both sides gives \begin {align*} \int -\frac {3}{2 y}d y &= x +c_{1}\\ -\frac {3 \ln \left (y \right )}{2}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-\frac {2 c_{1}}{3}-\frac {2 x}{3}}\\ &=c_{1} {\mathrm e}^{-\frac {2 x}{3}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-\frac {2 x}{3}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-\frac {2 x}{3}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y^{3}+3 y^{2} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {2 y}{3} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {2}{3} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\frac {2}{3}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\frac {2 x}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-\frac {2 x}{3}+c_{1}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
dsolve(2*y(x)^3+3*y(x)^2*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-\frac {2 x}{3}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 25
DSolve[2*y[x]^3+3*y[x]^2*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 0 \\ y(x)\to c_1 e^{-2 x/3} \\ y(x)\to 0 \\ \end{align*}