Internal problem ID [4112]
Internal file name [OUTPUT/3605_Sunday_June_05_2022_09_46_39_AM_89849220/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 875.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "linear", "quadrature", "separable", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {x {y^{\prime }}^{2}+\left (1-x^{2} y\right ) y^{\prime }-y x=0} \] The ode \begin {align*} x {y^{\prime }}^{2}+\left (1-x^{2} y\right ) y^{\prime }-y x = 0 \end {align*}
is factored to \begin {align*} \left (y^{\prime } x +1\right ) \left (y x -y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } x +1 = 0\tag {1} \\ y x -y^{\prime } = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { -\frac {1}{x}\,\mathop {\mathrm {d}x}}\\ &= -\ln \left (x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\ln \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\ln \left (x \right )+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\ln \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\ln \left (x \right )+c_{1} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= x y \end {align*}
Where \(f(x)=x\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= x \,d x\\ \int { \frac {1}{y} \,dy} &= \int {x \,d x}\\ \ln \left (y \right )&=\frac {x^{2}}{2}+c_{2}\\ y&={\mathrm e}^{\frac {x^{2}}{2}+c_{2}}\\ &=c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x {y^{\prime }}^{2}+\left (1-x^{2} y\right ) y^{\prime }-y x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {1}{x}, y^{\prime }=y x \right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {1}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\ln \left (x \right )+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=y x \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{\frac {x^{2}}{2}+c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\ln \left (x \right )+c_{1} , y={\mathrm e}^{\frac {x^{2}}{2}+c_{1}}\right \} \end {array} \]
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 <- quadrature successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 21
dsolve(x*diff(y(x),x)^2+(1-x^2*y(x))*diff(y(x),x)-x*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= {\mathrm e}^{\frac {x^{2}}{2}} c_{1} \\ y \left (x \right ) &= -\ln \left (x \right )+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 28
DSolve[x (y'[x])^2+(1-x^2 y[x])y'[x]-x y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{\frac {x^2}{2}} \\ y(x)\to -\log (x)+c_1 \\ \end{align*}