2.1.79 problem 78
Internal
problem
ID
[8217]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
78
Date
solved
:
Sunday, November 10, 2024 at 03:22:13 AM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
Solve
\begin{align*} \frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}}&=-x \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=3 \end{align*}
Solved as first order dAlembert ode
Time used: 0.576 (sec)
Let \(p=y^{\prime }\) the ode becomes
\begin{align*} \frac {p y}{1+\frac {\sqrt {p^{2}+1}}{2}} = -x \end{align*}
Solving for \(y\) from the above results in
\begin{align*}
\tag{1} y &= -\frac {x \left (2+\sqrt {p^{2}+1}\right )}{2 p} \\
\end{align*}
This has the form
\begin{align*} y=xf(p)+g(p)\tag {*} \end{align*}
Where \(f,g\) are functions of \(p=y'(x)\) . The above ode is dAlembert ode which is now solved.
Taking derivative of (*) w.r.t. \(x\) gives
\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}
Comparing the form \(y=x f + g\) to (1A) shows that
\begin{align*} f &= \frac {-2-\sqrt {p^{2}+1}}{2 p}\\ g &= 0 \end{align*}
Hence (2) becomes
\begin{align*} p -\frac {-2-\sqrt {p^{2}+1}}{2 p} = \left (-\frac {x}{2 \sqrt {p^{2}+1}}+\frac {x}{p^{2}}+\frac {x \sqrt {p^{2}+1}}{2 p^{2}}\right ) p^{\prime }\left (x \right )\tag {2A} \end{align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives
\begin{align*} p -\frac {-2-\sqrt {p^{2}+1}}{2 p} = 0 \end{align*}
No valid singular solutions found.
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\) . From eq. (2A). This results in
\begin{align*} p^{\prime }\left (x \right ) = \frac {p \left (x \right )-\frac {-2-\sqrt {p \left (x \right )^{2}+1}}{2 p \left (x \right )}}{-\frac {x}{2 \sqrt {p \left (x \right )^{2}+1}}+\frac {x}{p \left (x \right )^{2}}+\frac {x \sqrt {p \left (x \right )^{2}+1}}{2 p \left (x \right )^{2}}}\tag {3} \end{align*}
This ODE is now solved for \(p \left (x \right )\) . No inversion is needed. The ode \(p^{\prime }\left (x \right ) = \frac {\left (2 p \left (x \right )^{2}+\sqrt {p \left (x \right )^{2}+1}+2\right ) \sqrt {p \left (x \right )^{2}+1}\, p \left (x \right )}{x \left (1+2 \sqrt {p \left (x \right )^{2}+1}\right )}\) is separable as it can be
written as
\begin{align*} p^{\prime }\left (x \right )&= \frac {\left (2 p \left (x \right )^{2}+\sqrt {p \left (x \right )^{2}+1}+2\right ) \sqrt {p \left (x \right )^{2}+1}\, p \left (x \right )}{x \left (1+2 \sqrt {p \left (x \right )^{2}+1}\right )}\\ &= f(x) g(p) \end{align*}
Where
\begin{align*} f(x) &= \frac {1}{x}\\ g(p) &= \frac {\left (2 p^{2}+\sqrt {p^{2}+1}+2\right ) \sqrt {p^{2}+1}\, p}{1+2 \sqrt {p^{2}+1}} \end{align*}
Integrating gives
\begin{align*} \int { \frac {1}{g(p)} \,dp} &= \int { f(x) \,dx}\\ \int { \frac {1+2 \sqrt {p^{2}+1}}{\left (2 p^{2}+\sqrt {p^{2}+1}+2\right ) \sqrt {p^{2}+1}\, p}\,dp} &= \int { \frac {1}{x} \,dx}\\ \ln \left (\frac {p \left (x \right )}{\sqrt {p \left (x \right )^{2}+1}}\right )&=\ln \left (x \right )+c_1 \end{align*}
We now need to find the singular solutions, these are found by finding for what values \(g(p)\) is
zero, since we had to divide by this above. Solving \(g(p)=0\) or \(\frac {\left (2 p^{2}+\sqrt {p^{2}+1}+2\right ) \sqrt {p^{2}+1}\, p}{1+2 \sqrt {p^{2}+1}}=0\) for \(p \left (x \right )\) gives
\begin{align*} p \left (x \right )&=0\\ p \left (x \right )&=-i\\ p \left (x \right )&=i \end{align*}
Now we go over each such singular solution and check if it verifies the ode itself and
any initial conditions given. If it does not then the singular solution will not be
used.
Therefore the solutions found are
\begin{align*} \ln \left (\frac {p \left (x \right )}{\sqrt {p \left (x \right )^{2}+1}}\right ) = \ln \left (x \right )+c_1\\ p \left (x \right ) = 0\\ p \left (x \right ) = -i\\ p \left (x \right ) = i \end{align*}
Solving for \(p \left (x \right )\) gives
\begin{align*}
p \left (x \right ) &= \frac {{\mathrm e}^{c_1} x}{\sqrt {1-x^{2} {\mathrm e}^{2 c_1}}} \\
p \left (x \right ) &= -\frac {{\mathrm e}^{c_1} x}{\sqrt {1-x^{2} {\mathrm e}^{2 c_1}}} \\
\end{align*}
Substituing the above solution for \(p\) in (2A) gives
\begin{align*} y = -i x\\ y = i x\\ y = \frac {\left (-2-\sqrt {\frac {{\mathrm e}^{2 c_1} x^{2}}{1-x^{2} {\mathrm e}^{2 c_1}}+1}\right ) \sqrt {1-x^{2} {\mathrm e}^{2 c_1}}\, {\mathrm e}^{-c_1}}{2}\\ y = -\frac {\left (-2-\sqrt {\frac {{\mathrm e}^{2 c_1} x^{2}}{1-x^{2} {\mathrm e}^{2 c_1}}+1}\right ) \sqrt {1-x^{2} {\mathrm e}^{2 c_1}}\, {\mathrm e}^{-c_1}}{2}\\ \end{align*}
Solving for the constant of integration from initial conditions, the solution becomes
\begin{align*} y = -\left (-2-\sqrt {\frac {x^{2}}{-x^{2}+4}+1}\right ) \sqrt {1-\frac {x^{2}}{4}} \end{align*}
Solving for the constant of integration from initial conditions, the solution becomes
\begin{align*} y = -\left (-2-\sqrt {\frac {x^{2}}{-x^{2}+4}+1}\right ) \sqrt {1-\frac {x^{2}}{4}} \end{align*}
The solution
\[
y = -i x
\]
was found not to satisfy the ode or the IC. Hence it is removed.
The solution
\[
y = i x
\]
was found not to satisfy the ode or the IC. Hence it is removed.
Figure 2.163: Solutions plot
Summary of solutions found
\begin{align*}
y &= -\left (-2-\sqrt {\frac {x^{2}}{-x^{2}+4}+1}\right ) \sqrt {1-\frac {x^{2}}{4}} \\
y &= -\left (-2-\sqrt {\frac {x^{2}}{-x^{2}+4}+1}\right ) \sqrt {1-\frac {x^{2}}{4}} \\
\end{align*}
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {\left (\frac {d}{d x}y \left (x \right )\right ) y \left (x \right )}{1+\frac {\sqrt {1+\left (\frac {d}{d x}y \left (x \right )\right )^{2}}}{2}}=-x , y \left (0\right )=3\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d x}y \left (x \right )=-\frac {\left (4 y \left (x \right )-\sqrt {4 y \left (x \right )^{2}+3 x^{2}}\right ) x}{4 y \left (x \right )^{2}-x^{2}}, \frac {d}{d x}y \left (x \right )=-\frac {\left (4 y \left (x \right )+\sqrt {4 y \left (x \right )^{2}+3 x^{2}}\right ) x}{4 y \left (x \right )^{2}-x^{2}}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (4 y \left (x \right )-\sqrt {4 y \left (x \right )^{2}+3 x^{2}}\right ) x}{4 y \left (x \right )^{2}-x^{2}} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=3 \\ {} & {} & 0 \\ {} & \circ & \textrm {Solve for}\hspace {3pt} 0 \\ {} & {} & 0=0 \\ {} & \circ & \textrm {Substitute}\hspace {3pt} 0=0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & 0 \\ {} & \circ & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & 0 \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (4 y \left (x \right )+\sqrt {4 y \left (x \right )^{2}+3 x^{2}}\right ) x}{4 y \left (x \right )^{2}-x^{2}} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=3 \\ {} & {} & 0 \\ {} & \circ & \textrm {Solve for}\hspace {3pt} 0 \\ {} & {} & 0=0 \\ {} & \circ & \textrm {Substitute}\hspace {3pt} 0=0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & 0 \\ {} & \circ & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & 0 \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{0\right \} \end {array} \]
Maple trace
` Methods for first order ODEs:
-> Solving 1st order ODE of high degree, 1st attempt
trying 1st order WeierstrassP solution for high degree ODE
trying 1st order WeierstrassPPrime solution for high degree ODE
trying 1st order JacobiSN solution for high degree ODE
trying 1st order ODE linearizable_by_differentiation
trying differential order: 1; missing variables
trying dAlembert
<- dAlembert successful `
Maple dsolve solution
Solving time : 13.325
(sec)
Leaf size : 29
dsolve ([ diff ( y ( x ), x )* y ( x )/(1+1/2*(1+ diff ( y ( x ), x )^2)^(1/2)) = -x,
op ([ y (0) = 3])],y(x),singsol=all)
\begin{align*}
y &= -3+\sqrt {-x^{2}+36} \\
y &= 1+\sqrt {-x^{2}+4} \\
\end{align*}
Mathematica DSolve solution
Solving time : 0.5
(sec)
Leaf size : 35
DSolve [{ D [ y [ x ], x ]* y [ x ]/(1+1/2* Sqrt [1+( D [ y [ x ], x ])^2])==- x , y [0]==3},
y[x],x,IncludeSingularSolutions-> True ]
\begin{align*}
y(x)\to \sqrt {4-x^2}+1 \\
y(x)\to \sqrt {36-x^2}-3 \\
\end{align*}