Internal problem ID [5723]
Internal file name [OUTPUT/4971_Sunday_June_05_2022_03_15_35_PM_36475819/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: 10.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {2 y y^{\prime } x^{2}+y^{2}=2} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {y^{2}-2}{2 y \,x^{2}} \end {align*}
Where \(f(x)=-\frac {1}{2 x^{2}}\) and \(g(y)=\frac {y^{2}-2}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y^{2}-2}{y}} \,dy &= -\frac {1}{2 x^{2}} \,d x \\ \int { \frac {1}{\frac {y^{2}-2}{y}} \,dy} &= \int {-\frac {1}{2 x^{2}} \,d x} \\ \frac {\ln \left (y^{2}-2\right )}{2}&=\frac {1}{2 x}+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \sqrt {y^{2}-2} &= {\mathrm e}^{\frac {1}{2 x}+c_{1}} \end {align*}
Which simplifies to \begin {align*} \sqrt {y^{2}-2} &= c_{2} {\mathrm e}^{\frac {1}{2 x}} \end {align*}
The solution is \[ \sqrt {y^{2}-2} = c_{2} {\mathrm e}^{\frac {1}{2 x}+c_{1}} \]
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {y^{2}-2} &= c_{2} {\mathrm e}^{\frac {1}{2 x}+c_{1}} \\ \end{align*}
Verification of solutions
\[ \sqrt {y^{2}-2} = c_{2} {\mathrm e}^{\frac {1}{2 x}+c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y y^{\prime } x^{2}+y^{2}=2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{2}+2}{2 y x^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } y}{-y^{2}+2}=\frac {1}{2 x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } y}{-y^{2}+2}d x =\int \frac {1}{2 x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (y^{2}-2\right )}{2}=-\frac {1}{2 x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {2+{\mathrm e}^{-\frac {2 c_{1} x -1}{x}}}, y=-\sqrt {2+{\mathrm e}^{-\frac {2 c_{1} x -1}{x}}}\right \} \end {array} \]
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: 29
dsolve(2*x^2*y(x)*diff(y(x),x)+y(x)^2=2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +2} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.289 (sec). Leaf size: 70
DSolve[2*x*y[x]*y'[x]+y[x]^2==2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {2 x+e^{2 c_1}}}{\sqrt {x}} \\ y(x)\to \frac {\sqrt {2 x+e^{2 c_1}}}{\sqrt {x}} \\ y(x)\to -\sqrt {2} \\ y(x)\to \sqrt {2} \\ \end{align*}