Internal problem ID [4770]
Internal file name [OUTPUT/4263_Sunday_June_05_2022_12_49_44_PM_29245501/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: 10.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }+y \tanh \left (x \right )=2 \,{\mathrm e}^{x}} \]
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) &=\tanh \left (x \right )\\ q(x) &=2 \,{\mathrm e}^{x} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+y \tanh \left (x \right ) = 2 \,{\mathrm e}^{x} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \tanh \left (x \right )d x} \\ &= \cosh \left (x \right ) \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (2 \,{\mathrm e}^{x}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\cosh \left (x \right ) y\right ) &= \left (\cosh \left (x \right )\right ) \left (2 \,{\mathrm e}^{x}\right )\\ \mathrm {d} \left (\cosh \left (x \right ) y\right ) &= \left (2 \,{\mathrm e}^{x} \cosh \left (x \right )\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \cosh \left (x \right ) y &= \int {2 \,{\mathrm e}^{x} \cosh \left (x \right )\,\mathrm {d} x}\\ \cosh \left (x \right ) y &= \cosh \left (x \right )^{2}+\cosh \left (x \right ) \sinh \left (x \right )+x + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\cosh \left (x \right )\) results in \begin {align*} y &= \operatorname {sech}\left (x \right ) \left (\cosh \left (x \right )^{2}+\cosh \left (x \right ) \sinh \left (x \right )+x \right )+c_{1} \operatorname {sech}\left (x \right ) \end {align*}
which simplifies to \begin {align*} y &= \left (x +c_{1} \right ) \operatorname {sech}\left (x \right )+\cosh \left (x \right )+\sinh \left (x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (x +c_{1} \right ) \operatorname {sech}\left (x \right )+\cosh \left (x \right )+\sinh \left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = \left (x +c_{1} \right ) \operatorname {sech}\left (x \right )+\cosh \left (x \right )+\sinh \left (x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y \tanh \left (x \right )=2 \,{\mathrm e}^{x} \\ \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 }=-y \tanh \left (x \right )+2 \,{\mathrm e}^{x} \\ \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 }+y \tanh \left (x \right )=2 \,{\mathrm e}^{x} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+y \tanh \left (x \right )\right )=2 \mu \left (x \right ) {\mathrm e}^{x} \\ \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 }+y \tanh \left (x \right )\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 )=\mu \left (x \right ) \tanh \left (x \right ) \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\cosh \left (x \right ) \\ \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 2 \mu \left (x \right ) {\mathrm e}^{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int 2 \mu \left (x \right ) {\mathrm e}^{x}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int 2 \mu \left (x \right ) {\mathrm e}^{x}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\cosh \left (x \right ) \\ {} & {} & y=\frac {\int 2 \,{\mathrm e}^{x} \cosh \left (x \right )d x +c_{1}}{\cosh \left (x \right )} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\cosh \left (x \right )^{2}+\cosh \left (x \right ) \sinh \left (x \right )+x +c_{1}}{\cosh \left (x \right )} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\left (x +c_{1} \right ) \mathrm {sech}\left (x \right )+\cosh \left (x \right )+\sinh \left (x \right ) \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: 0.0 (sec). Leaf size: 15
dsolve(diff(y(x),x)+y(x)*tanh(x)=2*exp(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (x +c_{1} \right ) \operatorname {sech}\left (x \right )+\cosh \left (x \right )+\sinh \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.077 (sec). Leaf size: 29
DSolve[y'[x]+y[x]*Tanh[x]==2*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^x \left (2 x+e^{2 x}+c_1\right )}{e^{2 x}+1} \]