Internal problem ID [4235]
Internal file name [OUTPUT/3728_Sunday_June_05_2022_10_33_15_AM_54538900/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1004.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "bernoulli", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[[_homogeneous, `class A`], _rational, _Bernoulli]
\[ \boxed {4 x^{2} y^{2} {y^{\prime }}^{2}-\left (x^{2}+y^{2}\right )^{2}=0} \] The ode \begin {align*} 4 x^{2} y^{2} {y^{\prime }}^{2}-\left (x^{2}+y^{2}\right )^{2} = 0 \end {align*}
is factored to \begin {align*} \left (-2 y y^{\prime } x +y^{2}+x^{2}\right ) \left (2 y y^{\prime } x +y^{2}+x^{2}\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} -2 y y^{\prime } x +y^{2}+x^{2} = 0\tag {1} \\ 2 y y^{\prime } x +y^{2}+x^{2} = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} -2 u \left (x \right ) x^{2} \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right )+u \left (x \right )^{2} x^{2} = -x^{2} \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -\frac {u^{2}-1}{2 u x} \end {align*}
Where \(f(x)=-\frac {1}{2 x}\) and \(g(u)=\frac {u^{2}-1}{u}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {u^{2}-1}{u}} \,du &= -\frac {1}{2 x} \,d x \\ \int { \frac {1}{\frac {u^{2}-1}{u}} \,du} &= \int {-\frac {1}{2 x} \,d x} \\ \frac {\ln \left (u^{2}-1\right )}{2}&=-\frac {\ln \left (x \right )}{2}+c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} \sqrt {u^{2}-1} &= {\mathrm e}^{-\frac {\ln \left (x \right )}{2}+c_{2}} \end {align*}
Which simplifies to \begin {align*} \sqrt {u^{2}-1} &= \frac {c_{3}}{\sqrt {x}} \end {align*}
Which simplifies to \[ \sqrt {u \left (x \right )^{2}-1} = \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}} \] The solution is \[ \sqrt {u \left (x \right )^{2}-1} = \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}} \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \sqrt {\frac {y^{2}}{x^{2}}-1} = \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}}\\ \sqrt {\frac {y^{2}-x^{2}}{x^{2}}} = \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {\frac {y^{2}-x^{2}}{x^{2}}} &= \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}} \\ \end{align*}
Verification of solutions
\[ \sqrt {\frac {y^{2}-x^{2}}{x^{2}}} = \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {\frac {y^{2}-x^{2}}{x^{2}}} &= \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}} \\ \end{align*}
Verification of solutions
\[ \sqrt {\frac {y^{2}-x^{2}}{x^{2}}} = \frac {c_{3} {\mathrm e}^{c_{2}}}{\sqrt {x}} \] Verified OK.
Solving ODE (2) Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} 2 u \left (x \right ) x^{2} \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right )+u \left (x \right )^{2} x^{2} = -x^{2} \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -\frac {3 u^{2}+1}{2 u x} \end {align*}
Where \(f(x)=-\frac {1}{2 x}\) and \(g(u)=\frac {3 u^{2}+1}{u}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {3 u^{2}+1}{u}} \,du &= -\frac {1}{2 x} \,d x \\ \int { \frac {1}{\frac {3 u^{2}+1}{u}} \,du} &= \int {-\frac {1}{2 x} \,d x} \\ \frac {\ln \left (u^{2}+\frac {1}{3}\right )}{6}&=-\frac {\ln \left (x \right )}{2}+c_{5} \\ \end{align*} Raising both side to exponential gives \begin {align*} \frac {3^{\frac {2}{3}} \left (9 u^{2}+3\right )^{\frac {1}{6}}}{3} &= {\mathrm e}^{-\frac {\ln \left (x \right )}{2}+c_{5}} \end {align*}
Which simplifies to \begin {align*} \frac {3^{\frac {2}{3}} \left (9 u^{2}+3\right )^{\frac {1}{6}}}{3} &= \frac {c_{6}}{\sqrt {x}} \end {align*}
Which simplifies to \[ \frac {3^{\frac {2}{3}} \left (9 u \left (x \right )^{2}+3\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \] The solution is \[ \frac {3^{\frac {2}{3}} \left (9 u \left (x \right )^{2}+3\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \frac {3^{\frac {2}{3}} \left (\frac {9 y^{2}}{x^{2}}+3\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}}\\ \frac {3^{\frac {5}{6}} \left (\frac {3 y^{2}+x^{2}}{x^{2}}\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \frac {3^{\frac {5}{6}} \left (\frac {3 y^{2}+x^{2}}{x^{2}}\right )^{\frac {1}{6}}}{3} &= \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \\ \end{align*}
Verification of solutions
\[ \frac {3^{\frac {5}{6}} \left (\frac {3 y^{2}+x^{2}}{x^{2}}\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} \frac {3^{\frac {5}{6}} \left (\frac {3 y^{2}+x^{2}}{x^{2}}\right )^{\frac {1}{6}}}{3} &= \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \\ \end{align*}
Verification of solutions
\[ \frac {3^{\frac {5}{6}} \left (\frac {3 y^{2}+x^{2}}{x^{2}}\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 x^{2} y^{2} {y^{\prime }}^{2}-\left (x^{2}+y^{2}\right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {x^{2}+y^{2}}{2 y x}, y^{\prime }=\frac {x^{2}+y^{2}}{2 y x}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {x^{2}+y^{2}}{2 y x} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {x^{2}+y^{2}}{2 y x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
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: 69
dsolve(4*x^2*y(x)^2*diff(y(x),x)^2 = (x^2+y(x)^2)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {\left (x +c_{1} \right ) x} \\ y \left (x \right ) &= -\sqrt {\left (x +c_{1} \right ) x} \\ y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_{1} \right )}}{3 x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_{1} \right )}}{3 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.526 (sec). Leaf size: 97
DSolve[4 x^2 y[x]^2(y'[x])^2 ==(x^2+y[x]^2)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {x} \sqrt {x+c_1} \\ y(x)\to \sqrt {x} \sqrt {x+c_1} \\ y(x)\to -\frac {\sqrt {-x^3+3 c_1}}{\sqrt {3} \sqrt {x}} \\ y(x)\to \frac {\sqrt {-x^3+3 c_1}}{\sqrt {3} \sqrt {x}} \\ \end{align*}