Internal problem ID [2309]
Internal file name [OUTPUT/2309_Tuesday_February_27_2024_08_25_49_AM_20872197/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 35, page 157
Problem number: 37.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
\[ \boxed {y^{\prime \prime }-{y^{\prime }}^{2} \sin \left (x \right )=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
This is second order ode with missing dependent variable \(y\). Let \begin {align*} p(x) &= y^{\prime } \end {align*}
Then \begin {align*} p'(x) &= y^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} p^{\prime }\left (x \right )-p \left (x \right )^{2} \sin \left (x \right ) = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= p^{2} \sin \left (x \right ) \end {align*}
Where \(f(x)=\sin \left (x \right )\) and \(g(p)=p^{2}\). Integrating both sides gives \begin{align*} \frac {1}{p^{2}} \,dp &= \sin \left (x \right ) \,d x \\ \int { \frac {1}{p^{2}} \,dp} &= \int {\sin \left (x \right ) \,d x} \\ -\frac {1}{p}&=-\cos \left (x \right )+c_{1} \\ \end{align*} The solution is \[ -\frac {1}{p \left (x \right )}+\cos \left (x \right )-c_{1} = 0 \] Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(p={\frac {1}{2}}\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} -1-c_{1} = 0 \end {align*}
The solutions are \begin {align*} c_{1} = -1 \end {align*}
Trying the constant \begin {align*} c_{1} = -1 \end {align*}
Substituting \(c_{1}\) found above in the general solution gives \begin {align*} \frac {\cos \left (x \right ) p -1+p}{p} = 0 \end {align*}
The constant \(c_{1} = -1\) gives valid solution.
For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} \frac {y^{\prime } \cos \left (x \right )-1+y^{\prime }}{y^{\prime }} = 0 \end {align*}
Integrating both sides gives \begin {align*} y &= \int { \frac {1}{\cos \left (x \right )+1}\,\mathop {\mathrm {d}x}}\\ &= \tan \left (\frac {x}{2}\right )+c_{2} \end {align*}
Initial conditions are used to solve for \(c_{2}\). Substituting \(x=0\) and \(y=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 0 = c_{2} \end {align*}
The solutions are \begin {align*} c_{2} = 0 \end {align*}
Trying the constant \begin {align*} c_{2} = 0 \end {align*}
Substituting this in the general solution gives \begin {align*} y&=\tan \left (\frac {x}{2}\right ) \end {align*}
The constant \(c_{2} = 0\) gives valid solution.
Initial conditions are used to solve for the constants of integration.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \tan \left (\frac {x}{2}\right ) \\ \end{align*}
Verification of solutions
\[ y = \tan \left (\frac {x}{2}\right ) \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, diff(_b(_a), _a) = _b(_a)^2*sin(_a), _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✗ Solution by Maple
dsolve([diff(y(x),x$2)=diff(y(x),x)^2*sin(x),y(0) = 0, D(y)(0) = 1/2],y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 1.773 (sec). Leaf size: 6
DSolve[{y''[x]==y'[x]^2*Sin[x],{y[0]==0,y'[0]==1/2}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {Indeterminate} \]