Internal problem ID [4407]
Internal file name [OUTPUT/3900_Sunday_June_05_2022_11_37_40_AM_32836822/index.tex]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 7
Problem number: 2.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{y^{\prime }}^{2}=\frac {a^{2}}{x^{2}}} \] The ode \begin {align*} {y^{\prime }}^{2} = \frac {a^{2}}{x^{2}} \end {align*}
is factored to \begin {align*} \left (x y^{\prime }-a \right ) \left (x y^{\prime }+a \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} x y^{\prime }-a = 0\tag {1} \\ x y^{\prime }+a = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { \frac {a}{x}\,\mathop {\mathrm {d}x}}\\ &= a \ln \left (x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= a \ln \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = a \ln \left (x \right )+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= a \ln \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = a \ln \left (x \right )+c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} y &= \int { -\frac {a}{x}\,\mathop {\mathrm {d}x}}\\ &= -a \ln \left (x \right )+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -a \ln \left (x \right )+c_{2} \\ \end{align*}
Verification of solutions
\[ y = -a \ln \left (x \right )+c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -a \ln \left (x \right )+c_{2} \\ \end{align*}
Verification of solutions
\[ y = -a \ln \left (x \right )+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}=\frac {a^{2}}{x^{2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {a}{x}, y^{\prime }=-\frac {a}{x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {a}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {a}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=a \ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=a \ln \left (x \right )+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {a}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {a}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-a \ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-a \ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-a \ln \left (x \right )+c_{1} , y=a \ln \left (x \right )+c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 20
dsolve((diff(y(x),x))^2-a^2/x^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= a \ln \left (x \right )+c_{1} \\ y \left (x \right ) &= -a \ln \left (x \right )+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 24
DSolve[(y'[x])^2-a^2/x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -a \log (x)+c_1 \\ y(x)\to a \log (x)+c_1 \\ \end{align*}