Internal problem ID [8783]
Internal file name [OUTPUT/7718_Sunday_June_05_2022_11_48_47_PM_44854295/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 449.
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]
\[ \boxed {\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+2 y x y^{\prime }+y^{2}=0} \] The ode \begin {align*} \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+2 y x y^{\prime }+y^{2} = 0 \end {align*}
is factored to \begin {align*} \left (a y^{\prime }+x y^{\prime }+y\right ) \left (a y^{\prime }-x y^{\prime }-y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} a y^{\prime }+x y^{\prime }+y = 0\tag {1} \\ a y^{\prime }-x y^{\prime }-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}{-x +a} \end {align*}
Where \(f(x)=\frac {1}{-x +a}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= \frac {1}{-x +a} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {\frac {1}{-x +a} \,d x}\\ \ln \left (y \right )&=-\ln \left (-x +a \right )+c_{2}\\ y&={\mathrm e}^{-\ln \left (-x +a \right )+c_{2}}\\ &=\frac {c_{2}}{-x +a} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{-x +a} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{-x +a} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{-x +a} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{-x +a} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+2 y x y^{\prime }+y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {y}{-x +a}, y^{\prime }=-\frac {y}{a +x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {y}{-x +a} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=\frac {1}{-x +a} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \frac {1}{-x +a}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\ln \left (-x +a \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{-x +a} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {y}{a +x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {1}{a +x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\frac {1}{a +x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\ln \left (a +x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{a +x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {{\mathrm e}^{c_{1}}}{a +x}, y=\frac {{\mathrm e}^{c_{1}}}{-x +a}\right \} \end {array} \]
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}}{-x +a} \\ y \left (x \right ) &= \frac {c_{1}}{a +x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.049 (sec). Leaf size: 32
DSolve[y[x]^2 + 2*x*y[x]*y'[x] + (-a^2 + x^2)*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*}