Internal problem ID [4157]
Internal file name [OUTPUT/3650_Sunday_June_05_2022_10_01_14_AM_89327299/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 921.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "linear", "separable", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_separable]
\[ \boxed {\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-y^{2}=0} \] The ode \begin {align*} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-y^{2} = 0 \end {align*}
is factored to \begin {align*} \left (y^{\prime } a +y^{\prime } x +y\right ) \left (-y^{\prime } a +y^{\prime } x +y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } a +y^{\prime } x +y = 0\tag {1} \\ -y^{\prime } a +y^{\prime } x +y = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {y}{a +x} \end {align*}
Where \(f(x)=-\frac {1}{a +x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -\frac {1}{a +x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-\frac {1}{a +x} \,d x}\\ \ln \left (y \right )&=-\ln \left (a +x \right )+c_{1}\\ y&={\mathrm e}^{-\ln \left (a +x \right )+c_{1}}\\ &=\frac {c_{1}}{a +x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1}}{a +x} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1}}{a +x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1}}{a +x} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1}}{a +x} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y}{a -x} \end {align*}
Where \(f(x)=\frac {1}{a -x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= \frac {1}{a -x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {\frac {1}{a -x} \,d x}\\ \ln \left (y \right )&=-\ln \left (a -x \right )+c_{2}\\ y&={\mathrm e}^{-\ln \left (a -x \right )+c_{2}}\\ &=\frac {c_{2}}{a -x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{a -x} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{a -x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{a -x} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{a -x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {y}{a -x}, y^{\prime }=-\frac {y}{a +x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {y}{a -x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=\frac {1}{a -x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \frac {1}{a -x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\ln \left (a -x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{a -x} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {y}{a +x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {1}{a +x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\frac {1}{a +x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\ln \left (a +x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{a +x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {{\mathrm e}^{c_{1}}}{a -x}, y=\frac {{\mathrm e}^{c_{1}}}{a +x}\right \} \end {array} \]
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: 0.0 (sec). Leaf size: 23
dsolve((a^2-x^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {c_{1}}{a -x} \\ y \left (x \right ) &= \frac {c_{1}}{x +a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 32
DSolve[(a^2-x^2) (y'[x])^2-2 x y[x] y'[x]-y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1}{a-x} \\ y(x)\to \frac {c_1}{a+x} \\ y(x)\to 0 \\ \end{align*}