Internal problem ID [6093]
Internal file name [OUTPUT/5341_Sunday_June_05_2022_03_34_45_PM_92580117/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 6. Existence and uniqueness of solutions to systems and nth order equations.
Page 238
Problem number: 1(c).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y y^{\prime \prime }+4 {y^{\prime }}^{2}=0} \]
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(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} y p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+4 p \left (y \right )^{2} = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). In canonical form the ODE is \begin {align*} p' &= F(y,p)\\ &= f( y) g(p)\\ &= -\frac {4 p}{y} \end {align*}
Where \(f(y)=-\frac {4}{y}\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= -\frac {4}{y} \,d y\\ \int { \frac {1}{p} \,dp} &= \int {-\frac {4}{y} \,d y}\\ \ln \left (p \right )&=-4 \ln \left (y \right )+c_{1}\\ p&={\mathrm e}^{-4 \ln \left (y \right )+c_{1}}\\ &=\frac {c_{1}}{y^{4}} \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*} y^{\prime } = \frac {c_{1}}{y^{4}} \end {align*}
Integrating both sides gives \begin {align*} \int \frac {y^{4}}{c_{1}}d y &= x +c_{2}\\ \frac {y^{5}}{5 c_{1}}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}}\\ y_2&=\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}}\\ y_3&=\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}}\\ y_4&=\left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}}\\ y_5&=\left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \\ \tag{2} y &= \left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \\ \tag{3} y &= \left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \\ \tag{4} y &= \left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \\ \tag{5} y &= \left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \\ \end{align*}
Verification of solutions
\[ y = \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \] Verified OK.
\[ y = \left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \] Verified OK.
\[ y = \left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \] Verified OK.
\[ y = \left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \] Verified OK.
\[ y = \left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} c_{2} +5 c_{1} x \right )^{\frac {1}{5}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (\frac {d}{d x}y^{\prime }\right )+4 {y^{\prime }}^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Define new dependent variable}\hspace {3pt} u \\ {} & {} & u \left (x \right )=y^{\prime } \\ \bullet & {} & \textrm {Compute}\hspace {3pt} \frac {d}{d x}y^{\prime } \\ {} & {} & u^{\prime }\left (x \right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Use chain rule on the lhs}\hspace {3pt} \\ {} & {} & y^{\prime } \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Substitute in the definition of}\hspace {3pt} u \\ {} & {} & u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitutions}\hspace {3pt} y^{\prime }=u \left (y \right ),\frac {d}{d x}y^{\prime }=u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & y u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )+4 u \left (y \right )^{2}=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d y}u \left (y \right )=-\frac {4 u \left (y \right )}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d y}u \left (y \right )}{u \left (y \right )}=-\frac {4}{y} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} y \\ {} & {} & \int \frac {\frac {d}{d y}u \left (y \right )}{u \left (y \right )}d y =\int -\frac {4}{y}d y +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (u \left (y \right )\right )=-4 \ln \left (y \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {{\mathrm e}^{c_{1}}}{y^{4}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {{\mathrm e}^{c_{1}}}{y^{4}} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (y \right )=y^{\prime },y =y \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{c_{1}}}{y^{4}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{c_{1}}}{y^{4}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y^{4}={\mathrm e}^{c_{1}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } y^{4}d x =\int {\mathrm e}^{c_{1}}d x +c_{2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{5}}{5}=x \,{\mathrm e}^{c_{1}}+c_{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\left (5 x \,{\mathrm e}^{c_{1}}+5 c_{2} \right )^{\frac {1}{5}} \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 158
dsolve(y(x)*diff(y(x),x$2)+4*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \left (5 c_{1} x +5 c_{2} \right )^{\frac {1}{5}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left (5 c_{1} x +5 c_{2} \right )^{\frac {1}{5}}}{4} \\ y \left (x \right ) &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right ) \left (5 c_{1} x +5 c_{2} \right )^{\frac {1}{5}}}{4} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) \left (5 c_{1} x +5 c_{2} \right )^{\frac {1}{5}}}{4} \\ y \left (x \right ) &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) \left (5 c_{1} x +5 c_{2} \right )^{\frac {1}{5}}}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.178 (sec). Leaf size: 20
DSolve[y[x]*y''[x]+4*(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \sqrt [5]{5 x-c_1} \]