2.6.3 Problem 3 (eq 41)

Internal problem ID [18488]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 33. Problems at page 91
Problem number : 3 (eq 41)
Date solved : Thursday, March 13, 2025 at 12:06:35 PM
CAS classification : [[_2nd_order, _missing_x]]

Solve

\begin{align*} y^{\prime \prime }&=\frac {m \sqrt {1+{y^{\prime }}^{2}}}{k} \end{align*}

Solved as second order missing x ode

Time used: 2.582 (sec)

This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using

\begin{align*} y' &= p \end{align*}

Then

\begin{align*} y'' &= \frac {dp}{dx}\\ &= \frac {dp}{dy}\frac {dy}{dx}\\ &= p \frac {dp}{dy} \end{align*}

Hence the ode becomes

\begin{align*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right ) = \frac {m \sqrt {1+p \left (y \right )^{2}}}{k} \end{align*}

Which is now solved as first order ode for \(p(y)\).

Integrating gives

\begin{align*} \int \frac {k p}{m \sqrt {p^{2}+1}}d p &= dy\\ \frac {\sqrt {p^{2}+1}\, k}{m}&= y +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} \frac {m \sqrt {p^{2}+1}}{k p}&= 0 \end{align*}

for \(p\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} p = -i\\ p = i \end{align*}

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 {\sqrt {1+{y^{\prime }}^{2}}\, k}{m} = y+c_1 \end{align*}

Let \(p=y^{\prime }\) the ode becomes

\begin{align*} \frac {\sqrt {p^{2}+1}\, k}{m} = y +c_1 \end{align*}

Solving for \(y\) from the above results in

\begin{align*} \tag{1} y &= \frac {\sqrt {p^{2}+1}\, k -c_1 m}{m} \\ \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 &= 0\\ g &= \frac {\sqrt {p^{2}+1}\, k -c_1 m}{m} \end{align*}

Hence (2) becomes

\begin{equation} \tag{2A} p = \frac {k p p^{\prime }\left (x \right )}{\sqrt {p^{2}+1}\, m} \end{equation}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives

\begin{align*} p = 0 \end{align*}

Solving the above for \(p\) results in

\begin{align*} p_{1} &=0 \end{align*}

Substituting these in (1A) and keeping singular solution that verifies the ode gives

\begin{align*} y = \frac {-c_1 m +k}{m} \end{align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in

\begin{equation} \tag{3} p^{\prime }\left (x \right ) = \frac {\sqrt {p \left (x \right )^{2}+1}\, m}{k} \end{equation}

This ODE is now solved for \(p \left (x \right )\). No inversion is needed.

Integrating gives

\begin{align*} \int \frac {k}{\sqrt {p^{2}+1}\, m}d p &= dx\\ \frac {k \,\operatorname {arcsinh}\left (p \right )}{m}&= x +c_2 \end{align*}

Singular solutions are found by solving

\begin{align*} \frac {\sqrt {p^{2}+1}\, m}{k}&= 0 \end{align*}

for \(p \left (x \right )\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} p \left (x \right ) = -i\\ p \left (x \right ) = i \end{align*}

Substituing the above solution for \(p\) in (2A) gives

\begin{align*} y &= \frac {\sqrt {\sinh \left (\frac {m \left (x +c_2 \right )}{k}\right )^{2}+1}\, k -c_1 m}{m} \\ y &= -c_1 \\ y &= -c_1 \\ \end{align*}

For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = -i \end{align*}

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {-i\, dx}\\ y &= -i x + c_3 \end{align*}

For solution (3) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = i \end{align*}

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {i\, dx}\\ y &= i x + c_4 \end{align*}

Will add steps showing solving for IC soon.

The solution

\[ y = \frac {-c_1 m +k}{m} \]

was found not to satisfy the ode or the IC. Hence it is removed. The solution

\[ y = -c_1 \]

was found not to satisfy the ode or the IC. Hence it is removed.

Summary of solutions found

\begin{align*} y &= \frac {\sqrt {\sinh \left (\frac {m \left (x +c_2 \right )}{k}\right )^{2}+1}\, k -c_1 m}{m} \\ y &= -i x +c_3 \\ y &= i x +c_4 \\ \end{align*}

Solved as second order missing y ode

Time used: 0.298 (sec)

This is second order ode with missing dependent variable \(y\). Let

\begin{align*} u(x) &= y^{\prime } \end{align*}

Then

\begin{align*} u'(x) &= y^{\prime \prime } \end{align*}

Hence the ode becomes

\begin{align*} u^{\prime }\left (x \right )-\frac {m \sqrt {1+u \left (x \right )^{2}}}{k} = 0 \end{align*}

Which is now solved for \(u(x)\) as first order ode.

Integrating gives

\begin{align*} \int \frac {k}{m \sqrt {u^{2}+1}}d u &= dx\\ \frac {k \,\operatorname {arcsinh}\left (u \right )}{m}&= x +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} \frac {m \sqrt {u^{2}+1}}{k}&= 0 \end{align*}

for \(u \left (x \right )\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} u \left (x \right ) = -i\\ u \left (x \right ) = i \end{align*}

In summary, these are the solution found for \(u(x)\)

\begin{align*} \frac {k \,\operatorname {arcsinh}\left (u \left (x \right )\right )}{m} &= x +c_1 \\ u \left (x \right ) &= -i \\ u \left (x \right ) &= i \\ \end{align*}

For solution \(\frac {k \,\operatorname {arcsinh}\left (u \left (x \right )\right )}{m} = x +c_1\), since \(u=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} \frac {k \,\operatorname {arcsinh}\left (y^{\prime }\right )}{m} = x +c_1 \end{align*}

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {\sinh \left (\frac {m \left (x +c_1 \right )}{k}\right )\, dx}\\ y &= \frac {k \cosh \left (\frac {m x}{k}+\frac {m c_1}{k}\right )}{m} + c_2 \end{align*}

\begin{align*} y&= \frac {k \cosh \left (\frac {m \left (x +c_1 \right )}{k}\right )}{m}+c_2 \end{align*}

For solution \(u \left (x \right ) = -i\), since \(u=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = -i \end{align*}

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {-i\, dx}\\ y &= -i x + c_3 \end{align*}

For solution \(u \left (x \right ) = i\), since \(u=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = i \end{align*}

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {i\, dx}\\ y &= i x + c_4 \end{align*}

In summary, these are the solution found for \((y)\)

\begin{align*} y &= \frac {k \cosh \left (\frac {m \left (x +c_1 \right )}{k}\right )}{m}+c_2 \\ y &= -i x +c_3 \\ y &= i x +c_4 \\ \end{align*}

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} y &= -i x +c_3 \\ y &= i x +c_4 \\ y &= \frac {k \cosh \left (\frac {m \left (x +c_1 \right )}{k}\right )}{m}+c_2 \\ \end{align*}

✓ Maple. Time used: 0.284 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x) = m/k*(1+diff(y(x),x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \frac {k \cosh \left (\frac {m \left (x +c_{1} \right )}{k}\right )}{m}+c_{2} \\ \end{align*}

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 
-> Calling odsolve with the ODE`, diff(diff(diff(y(x), x), x), x)-m^2*(diff(y(x), x))/k^2, y(x)`   *** Sublevel 2 *** 
   Methods for third order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   checking if the LODE has constant coefficients 
   <- constant coefficients successful 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
`, `-> Computing symmetries using: way = 3 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = m*(_b(_a)^2+1)^(1/2)/k, _b(_a), HINT = [[1, 0]]`   *** Sublevel 2 *** 
   symmetry methods on request 
`, `1st order, trying reduction of order with given symmetries:`[1, 0]
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=\frac {m \sqrt {1+{y^{\prime }}^{2}}}{k} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=\frac {m \sqrt {1+u \left (x \right )^{2}}}{k} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=\frac {m \sqrt {1+u \left (x \right )^{2}}}{k} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{\sqrt {1+u \left (x \right )^{2}}}=\frac {m}{k} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{\sqrt {1+u \left (x \right )^{2}}}d x =\int \frac {m}{k}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \mathrm {arcsinh}\left (u \left (x \right )\right )=\frac {m x}{k}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\sinh \left (\frac {\mathit {C1} k +x m}{k}\right ) \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\sinh \left (\frac {\mathit {C1} k +x m}{k}\right ) \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=\sinh \left (\frac {\mathit {C1} k +x m}{k}\right ) \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \sinh \left (\frac {\mathit {C1} k +x m}{k}\right )d x +\mathit {C2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\frac {k \cosh \left (\frac {m x}{k}+\mathit {C1} \right )}{m}+\mathit {C2} \end {array} \]

✓ Mathematica. Time used: 0.357 (sec). Leaf size: 23

ode=D[y[x],{x,2}]==m/k*Sqrt[1+D[y[x],x]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {k \cosh \left (\frac {m x}{k}+c_1\right )}{m}+c_2 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
m = symbols("m") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - m*sqrt(Derivative(y(x), x)**2 + 1)/k,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out