Internal problem ID [12122]
Internal file name [OUTPUT/10775_Tuesday_September_12_2023_08_51_44_AM_49389396/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 12.
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]
\[ \boxed {{y^{\prime }}^{2}-9 y^{4}=0} \] The ode \begin {align*} {y^{\prime }}^{2}-9 y^{4} = 0 \end {align*}
is factored to \begin {align*} \left (3 y^{2}-y^{\prime }\right ) \left (3 y^{2}+y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} 3 y^{2}-y^{\prime } = 0\tag {1} \\ 3 y^{2}+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}{3 y^{2}}d y &= x +c_{1}\\ -\frac {1}{3 y}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {1}{3 \left (x +c_{1} \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{3 \left (x +c_{1} \right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{3 \left (x +c_{1} \right )} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{3 \left (x +c_{1} \right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{3 \left (x +c_{1} \right )} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} \int -\frac {1}{3 y^{2}}d y &= x +c_{2}\\ \frac {1}{3 y}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{3 c_{2} +3 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{3 c_{2} +3 x} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{3 c_{2} +3 x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{3 c_{2} +3 x} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{3 c_{2} +3 x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-9 y^{4}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-3 y^{2}, y^{\prime }=3 y^{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-3 y^{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=-3 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d x =\int \left (-3\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=-3 x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{-3 x +c_{1}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=3 y^{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=3 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d x =\int 3d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=3 x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{3 x +c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\frac {1}{-3 x +c_{1}}, y=-\frac {1}{3 x +c_{1}}\right \} \end {array} \]
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: 0.0 (sec). Leaf size: 21
dsolve(diff(y(x),x)^2=9*y(x)^4,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{c_{1} -3 x} \\ y \left (x \right ) &= \frac {1}{3 x +c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.263 (sec). Leaf size: 34
DSolve[y'[x]^2==9*y[x]^4,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{3 x+c_1} \\ y(x)\to \frac {1}{3 x-c_1} \\ y(x)\to 0 \\ \end{align*}