Internal problem ID [3999]
Internal file name [OUTPUT/3492_Sunday_June_05_2022_09_28_05_AM_69444829/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 758.
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]
\[ \boxed {{y^{\prime }}^{2}-a^{2} y^{2}=0} \] The ode \begin {align*} {y^{\prime }}^{2}-a^{2} y^{2} = 0 \end {align*}
is factored to \begin {align*} \left (-a y+y^{\prime }\right ) \left (a y+y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} -a y+y^{\prime } = 0\tag {1} \\ a y+y^{\prime } = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} \int \frac {1}{a y}d y &= x +c_{1}\\ \frac {\ln \left (y \right )}{a}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{a c_{1} +a x}\\ &=c_{1} {\mathrm e}^{a x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{a x} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{a x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{a x} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{a x} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} \int -\frac {1}{a y}d y &= x +c_{2}\\ -\frac {\ln \left (y \right )}{a}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-a c_{2} -a x}\\ &=\frac {{\mathrm e}^{-a x}}{c_{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-a x}}{c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-a x}}{c_{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-a x}}{c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-a x}}{c_{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-a^{2} y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=a y, y^{\prime }=-a y\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=a y \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=a \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int a d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=a x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{a x +c_{1}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-a y \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-a \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -a d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-a x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-a x +c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y={\mathrm e}^{-a x +c_{1}}, y={\mathrm e}^{a x +c_{1}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 20
dsolve(diff(y(x),x)^2 = a^2*y(x)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} {\mathrm e}^{a x} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-a x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.069 (sec). Leaf size: 31
DSolve[(y'[x])^2==a^2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{-a x} \\ y(x)\to c_1 e^{a x} \\ y(x)\to 0 \\ \end{align*}