Internal problem ID [4766]
Internal file name [OUTPUT/4259_Sunday_June_05_2022_12_49_09_PM_4922600/index.tex]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 3. Linear First-Order Equations. page
403
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
[_linear]
\[ \boxed {y^{\prime }+\frac {y}{\sqrt {x^{2}+1}}=\frac {1}{x +\sqrt {x^{2}+1}}} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=\frac {1}{\sqrt {x^{2}+1}}\\ q(x) &=\frac {1}{x +\sqrt {x^{2}+1}} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {y}{\sqrt {x^{2}+1}} = \frac {1}{x +\sqrt {x^{2}+1}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {1}{\sqrt {x^{2}+1}}d x} \\ &= x +\sqrt {x^{2}+1} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\frac {1}{x +\sqrt {x^{2}+1}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\left (x +\sqrt {x^{2}+1}\right ) y\right ) &= \left (x +\sqrt {x^{2}+1}\right ) \left (\frac {1}{x +\sqrt {x^{2}+1}}\right )\\ \mathrm {d} \left (\left (x +\sqrt {x^{2}+1}\right ) y\right ) &= \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \left (x +\sqrt {x^{2}+1}\right ) y &= \int {\mathrm {d} x}\\ \left (x +\sqrt {x^{2}+1}\right ) y &= x + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =x +\sqrt {x^{2}+1}\) results in \begin {align*} y &= \frac {x}{x +\sqrt {x^{2}+1}}+\frac {c_{1}}{x +\sqrt {x^{2}+1}} \end {align*}
which simplifies to \begin {align*} y &= \frac {x +c_{1}}{x +\sqrt {x^{2}+1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x +c_{1}}{x +\sqrt {x^{2}+1}} \\ \end{align*}
Verification of solutions
\[ y = \frac {x +c_{1}}{x +\sqrt {x^{2}+1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {y}{\sqrt {x^{2}+1}}=\frac {1}{x +\sqrt {x^{2}+1}} \\ \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}{\sqrt {x^{2}+1}}+\frac {1}{x +\sqrt {x^{2}+1}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {y}{\sqrt {x^{2}+1}}=\frac {1}{x +\sqrt {x^{2}+1}} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+\frac {y}{\sqrt {x^{2}+1}}\right )=\frac {\mu \left (x \right )}{x +\sqrt {x^{2}+1}} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (y \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+\frac {y}{\sqrt {x^{2}+1}}\right )=y^{\prime } \mu \left (x \right )+y \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=\frac {\mu \left (x \right )}{\sqrt {x^{2}+1}} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\frac {x \sqrt {x^{2}+1}+x^{2}+1}{\sqrt {x^{2}+1}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (y \mu \left (x \right )\right )\right )d x =\int \frac {\mu \left (x \right )}{x +\sqrt {x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \frac {\mu \left (x \right )}{x +\sqrt {x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \frac {\mu \left (x \right )}{x +\sqrt {x^{2}+1}}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\frac {x \sqrt {x^{2}+1}+x^{2}+1}{\sqrt {x^{2}+1}} \\ {} & {} & y=\frac {\sqrt {x^{2}+1}\, \left (\int \frac {x \sqrt {x^{2}+1}+x^{2}+1}{\sqrt {x^{2}+1}\, \left (x +\sqrt {x^{2}+1}\right )}d x +c_{1} \right )}{x \sqrt {x^{2}+1}+x^{2}+1} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\sqrt {x^{2}+1}\, \left (x +c_{1} \right )}{x \sqrt {x^{2}+1}+x^{2}+1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 19
dsolve(diff(y(x),x)+y(x)/sqrt(x^2+1)=1/(x+sqrt(x^2+1)),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x +c_{1}}{x +\sqrt {x^{2}+1}} \]
✓ Solution by Mathematica
Time used: 0.121 (sec). Leaf size: 23
DSolve[y'[x]+y[x]/Sqrt[x^2+1]==1/(x+Sqrt[x^2+1]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (\sqrt {x^2+1}-x\right ) (x+c_1) \]