Internal problem ID [5715]
Internal file name [OUTPUT/4963_Sunday_June_05_2022_03_15_20_PM_92366259/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems.
page 7
Problem number: 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {x^{2}}{y \left (x^{3}+1\right )}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x^{2}}{y \left (x^{3}+1\right )} \end {align*}
Where \(f(x)=\frac {x^{2}}{x^{3}+1}\) and \(g(y)=\frac {1}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y}} \,dy &= \frac {x^{2}}{x^{3}+1} \,d x \\ \int { \frac {1}{\frac {1}{y}} \,dy} &= \int {\frac {x^{2}}{x^{3}+1} \,d x} \\ \frac {y^{2}}{2}&=\frac {\ln \left (x^{3}+1\right )}{3}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \\ y &= -\frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \\ \tag{2} y &= -\frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \] Verified OK.
\[ y = -\frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {x^{2}}{y \left (x^{3}+1\right )}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}}{y \left (x^{3}+1\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=\frac {x^{2}}{x^{3}+1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int \frac {x^{2}}{x^{3}+1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=\frac {\ln \left (x^{3}+1\right )}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3}, y=\frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(diff(y(x),x)=x^2/(y(x)*(1+x^3)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6 \ln \left (x^{3}+1\right )+9 c_{1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6 \ln \left (x^{3}+1\right )+9 c_{1}}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.091 (sec). Leaf size: 56
DSolve[y'[x]==x^2/(y[x]*(1+x^3)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {\frac {2}{3}} \sqrt {\log \left (x^3+1\right )+3 c_1} \\ y(x)\to \sqrt {\frac {2}{3}} \sqrt {\log \left (x^3+1\right )+3 c_1} \\ \end{align*}