Internal problem ID [4203]
Internal file name [OUTPUT/3696_Sunday_June_05_2022_10_17_19_AM_9607831/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 970.
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]
\[ \boxed {y^{2} {y^{\prime }}^{2}=a^{2}} \] The ode \begin {align*} y^{2} {y^{\prime }}^{2} = a^{2} \end {align*}
is factored to \begin {align*} \left (y^{\prime } y-a \right ) \left (y^{\prime } y+a \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } y-a = 0\tag {1} \\ y^{\prime } y+a = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} \int \frac {y}{a}d y &= x +c_{1}\\ \frac {y^{2}}{2 a}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {2 a c_{1} +2 a x}\\ y_2&=-\sqrt {2 a c_{1} +2 a x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {2 a c_{1} +2 a x} \\ \tag{2} y &= -\sqrt {2 a c_{1} +2 a x} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {2 a c_{1} +2 a x} \] Verified OK.
\[ y = -\sqrt {2 a c_{1} +2 a x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {2 a c_{1} +2 a x} \\ \tag{2} y &= -\sqrt {2 a c_{1} +2 a x} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {2 a c_{1} +2 a x} \] Verified OK.
\[ y = -\sqrt {2 a c_{1} +2 a x} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} \int -\frac {y}{a}d y &= x +c_{2}\\ -\frac {y^{2}}{2 a}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {-2 a c_{2} -2 a x}\\ y_2&=-\sqrt {-2 a c_{2} -2 a x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-2 a c_{2} -2 a x} \\ \tag{2} y &= -\sqrt {-2 a c_{2} -2 a x} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-2 a c_{2} -2 a x} \] Verified OK.
\[ y = -\sqrt {-2 a c_{2} -2 a x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-2 a c_{2} -2 a x} \\ \tag{2} y &= -\sqrt {-2 a c_{2} -2 a x} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-2 a c_{2} -2 a x} \] Verified OK.
\[ y = -\sqrt {-2 a c_{2} -2 a x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{2} {y^{\prime }}^{2}=a^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {a}{y}, y^{\prime }=-\frac {a}{y}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {a}{y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=a \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int a d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=a x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {2 a x +2 c_{1}}, y=-\sqrt {2 a x +2 c_{1}}\right \} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {a}{y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=-a \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int -a d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-a x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-2 a x +2 c_{1}}, y=-\sqrt {-2 a x +2 c_{1}}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\left \{y=\sqrt {-2 a x +2 c_{1}}, y=-\sqrt {-2 a x +2 c_{1}}\right \}, \left \{y=\sqrt {2 a x +2 c_{1}}, y=-\sqrt {2 a x +2 c_{1}}\right \}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 49
dsolve(y(x)^2*diff(y(x),x)^2 = a^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {2 a x +c_{1}} \\ y \left (x \right ) &= -\sqrt {2 a x +c_{1}} \\ y \left (x \right ) &= \sqrt {-2 a x +c_{1}} \\ y \left (x \right ) &= -\sqrt {-2 a x +c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.285 (sec). Leaf size: 85
DSolve[y[x]^2(y'[x])^2==a^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {-a x+c_1} \\ y(x)\to \sqrt {2} \sqrt {-a x+c_1} \\ y(x)\to -\sqrt {2} \sqrt {a x+c_1} \\ y(x)\to \sqrt {2} \sqrt {a x+c_1} \\ \end{align*}