Internal problem ID [3143]
Internal file name [OUTPUT/2635_Sunday_June_05_2022_08_37_48_AM_26764279/index.tex]
Book: An introduction to the solution and applications of differential equations, J.W. Searl,
1966
Section: Chapter 4, Ex. 4.2
Problem number: 4.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\sqrt {x^{2}+1}\, y^{\prime }+\sqrt {1+y^{2}}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {\sqrt {y^{2}+1}}{\sqrt {x^{2}+1}} \end {align*}
Where \(f(x)=-\frac {1}{\sqrt {x^{2}+1}}\) and \(g(y)=\sqrt {y^{2}+1}\). Integrating both sides gives \begin{align*} \frac {1}{\sqrt {y^{2}+1}} \,dy &= -\frac {1}{\sqrt {x^{2}+1}} \,d x \\ \int { \frac {1}{\sqrt {y^{2}+1}} \,dy} &= \int {-\frac {1}{\sqrt {x^{2}+1}} \,d x} \\ \operatorname {arcsinh}\left (y \right )&=-\operatorname {arcsinh}\left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \sinh \left (-\operatorname {arcsinh}\left (x \right )+c_{1} \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sinh \left (-\operatorname {arcsinh}\left (x \right )+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \sinh \left (-\operatorname {arcsinh}\left (x \right )+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sqrt {x^{2}+1}\, y^{\prime }+\sqrt {1+y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\sqrt {1+y^{2}}}{\sqrt {x^{2}+1}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {1+y^{2}}}=-\frac {1}{\sqrt {x^{2}+1}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {1+y^{2}}}d x =\int -\frac {1}{\sqrt {x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \mathrm {arcsinh}\left (y\right )=-\mathrm {arcsinh}\left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\sinh \left (-\mathrm {arcsinh}\left (x \right )+c_{1} \right ) \end {array} \]
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: 0.0 (sec). Leaf size: 11
dsolve(sqrt(1+x^2)*diff(y(x),x)+sqrt(1+y(x)^2)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\sinh \left (\operatorname {arcsinh}\left (x \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.349 (sec). Leaf size: 59
DSolve[Sqrt[1+x^2]*y'[x]+Sqrt[1+y[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-c_1} \left (\left (-1+e^{2 c_1}\right ) \sqrt {x^2+1}-\left (1+e^{2 c_1}\right ) x\right ) \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}