Internal problem ID [5719]
Internal file name [OUTPUT/4967_Sunday_June_05_2022_03_15_26_PM_94014602/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: 6.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x y y^{\prime }-\sqrt {1+y^{2}}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {\sqrt {y^{2}+1}}{x y} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(y)=\frac {\sqrt {y^{2}+1}}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {\sqrt {y^{2}+1}}{y}} \,dy &= \frac {1}{x} \,d x \\ \int { \frac {1}{\frac {\sqrt {y^{2}+1}}{y}} \,dy} &= \int {\frac {1}{x} \,d x} \\ \sqrt {y^{2}+1}&=\ln \left (x \right )+c_{1} \\ \end{align*} The solution is \[ \sqrt {1+y^{2}}-\ln \left (x \right )-c_{1} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {1+y^{2}}-\ln \left (x \right )-c_{1} &= 0 \\ \end{align*}
Verification of solutions
\[ \sqrt {1+y^{2}}-\ln \left (x \right )-c_{1} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y y^{\prime }-\sqrt {1+y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {1+y^{2}}}{x y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y y^{\prime }}{\sqrt {1+y^{2}}}=\frac {1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y y^{\prime }}{\sqrt {1+y^{2}}}d x =\int \frac {1}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \sqrt {1+y^{2}}=\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-1+c_{1}^{2}+2 c_{1} \ln \left (x \right )+\ln \left (x \right )^{2}}, y=-\sqrt {-1+c_{1}^{2}+2 c_{1} \ln \left (x \right )+\ln \left (x \right )^{2}}\right \} \end {array} \]
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: 17
dsolve(x*y(x)*diff(y(x),x)=sqrt(1+y(x)^2),y(x), singsol=all)
\[ \ln \left (x \right )-\sqrt {1+y \left (x \right )^{2}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.229 (sec). Leaf size: 65
DSolve[x*y[x]*y'[x]==Sqrt[1+y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {\log ^2(x)+2 c_1 \log (x)-1+c_1{}^2} \\ y(x)\to \sqrt {\log ^2(x)+2 c_1 \log (x)-1+c_1{}^2} \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}