Internal problem ID [4062]
Internal file name [OUTPUT/3555_Sunday_June_05_2022_09_38_19_AM_37848521/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 823.
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
[_separable]
\[ \boxed {{y^{\prime }}^{2}-\left (x -y\right ) y y^{\prime }-x y^{3}=0} \] The ode \begin {align*} {y^{\prime }}^{2}-\left (x -y\right ) y y^{\prime }-x y^{3} = 0 \end {align*}
is factored to \begin {align*} \left (y^{2}+y^{\prime }\right ) \left (x y-y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{2}+y^{\prime } = 0\tag {1} \\ x y-y^{\prime } = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} \int -\frac {1}{y^{2}}d y &= x +c_{1}\\ \frac {1}{y}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{x +c_{1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{x +c_{1}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{x +c_{1}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{x +c_{1}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{x +c_{1}} \] 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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-\left (x -y\right ) y y^{\prime }-x y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=x y, y^{\prime }=-y^{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=x y \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{\frac {x^{2}}{2}+c_{1}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-y^{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=-1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d x =\int \left (-1\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=-x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{-x +c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\frac {1}{-x +c_{1}}, y={\mathrm e}^{\frac {x^{2}}{2}+c_{1}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli 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: 20
dsolve(diff(y(x),x)^2-(x-y(x))*y(x)*diff(y(x),x)-x*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{c_{1} +x} \\ y \left (x \right ) &= {\mathrm e}^{\frac {x^{2}}{2}} c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.136 (sec). Leaf size: 34
DSolve[(y'[x])^2-(x-y[x])y[x] y'[x]-x y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{x-c_1} \\ y(x)\to c_1 e^{\frac {x^2}{2}} \\ y(x)\to 0 \\ \end{align*}