Internal problem ID [5718]
Internal file name [OUTPUT/4966_Sunday_June_05_2022_03_15_25_PM_87721000/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: 5.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {x^{2}}{1+y^{2}}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x^{2}}{y^{2}+1} \end {align*}
Where \(f(x)=x^{2}\) and \(g(y)=\frac {1}{y^{2}+1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y^{2}+1}} \,dy &= x^{2} \,d x \\ \int { \frac {1}{\frac {1}{y^{2}+1}} \,dy} &= \int {x^{2} \,d x} \\ \frac {1}{3} y^{3}+y&=\frac {x^{3}}{3}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y &= -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ y &= -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ \tag{2} y &= -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \tag{3} y &= -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \] Verified OK.
\[ y = -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.
\[ y = -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {x^{2}}{1+y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}}{1+y^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } \left (1+y^{2}\right )=x^{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } \left (1+y^{2}\right )d x =\int x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{3}}{3}+y=\frac {x^{3}}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \end {array} \]
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: 0.015 (sec). Leaf size: 268
dsolve(diff(y(x),x)=x^2/(1+y(x)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}-4}{2 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}+4 i \sqrt {3}-4}{4 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}} \sqrt {3}+4 i \sqrt {3}-\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}+4}{4 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.179 (sec). Leaf size: 307
DSolve[y'[x]==x^2/(1+y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2+\sqrt [3]{2} \left (x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\ \end{align*}