Internal problem ID [5729]
Internal file name [OUTPUT/4977_Sunday_June_05_2022_03_15_44_PM_75195143/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: 16.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\left (y+2 y^{3}\right ) y^{\prime }=-2 x^{3}-x} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {x \left (2 x^{2}+1\right )}{2 y^{3}+y} \end {align*}
Where \(f(x)=-x \left (2 x^{2}+1\right )\) and \(g(y)=\frac {1}{2 y^{3}+y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{2 y^{3}+y}} \,dy &= -x \left (2 x^{2}+1\right ) \,d x \\ \int { \frac {1}{\frac {1}{2 y^{3}+y}} \,dy} &= \int {-x \left (2 x^{2}+1\right ) \,d x} \\ \frac {y^{2} \left (y^{2}+1\right )}{2}&=-\frac {\left (2 x^{2}+1\right )^{2}}{8}+c_{1} \\ \end{align*} The solution is \[ \frac {y^{2} \left (1+y^{2}\right )}{2}+\frac {\left (2 x^{2}+1\right )^{2}}{8}-c_{1} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} \frac {y^{2} \left (1+y^{2}\right )}{2}+\frac {\left (2 x^{2}+1\right )^{2}}{8}-c_{1} &= 0 \\ \end{align*}
Verification of solutions
\[ \frac {y^{2} \left (1+y^{2}\right )}{2}+\frac {\left (2 x^{2}+1\right )^{2}}{8}-c_{1} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y+2 y^{3}\right ) y^{\prime }=-2 x^{3}-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (y+2 y^{3}\right ) y^{\prime }d x =\int \left (-2 x^{3}-x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\left (2 y^{2}+1\right )^{2}}{8}=-\frac {\left (2 x^{2}+1\right )^{2}}{8}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 113
dsolve((x+2*x^3)+(y(x)+2*y(x)^3)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2-2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2-2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {-2+2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2+2 \sqrt {-4 x^{4}-4 x^{2}-8 c_{1} -1}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.086 (sec). Leaf size: 151
DSolve[(x+2*x^3)+(y[x]+2*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-1-\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1-\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-1+\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1+\sqrt {-4 x^4-4 x^2+1+8 c_1}}}{\sqrt {2}} \\ \end{align*}