Internal problem ID [4091]
Internal file name [OUTPUT/3584_Sunday_June_05_2022_09_43_20_AM_2007886/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 852.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {x {y^{\prime }}^{2}-\left (x^{2}+1\right ) y^{\prime }=-x} \] The ode \begin {align*} x {y^{\prime }}^{2}-\left (x^{2}+1\right ) y^{\prime } = -x \end {align*}
is factored to \begin {align*} \left (y^{\prime } x -1\right ) \left (-x +y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } x -1 = 0\tag {1} \\ -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) Integrating both sides gives \begin {align*} y &= \int { x\,\mathop {\mathrm {d}x}}\\ &= \frac {x^{2}}{2}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2}}{2}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{2}}{2}+c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2}}{2}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{2}}{2}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x {y^{\prime }}^{2}-\left (x^{2}+1\right ) y^{\prime }=-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=x , y^{\prime }=\frac {1}{x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x^{2}}{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {x^{2}}{2}+c_{1} , y=\ln \left (x \right )+c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 18
dsolve(x*diff(y(x),x)^2-(x^2+1)*diff(y(x),x)+x = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \ln \left (x \right )+c_{1} \\ y \left (x \right ) &= \frac {x^{2}}{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 24
DSolve[x (y'[x])^2-(1+x^2)y'[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^2}{2}+c_1 \\ y(x)\to \log (x)+c_1 \\ \end{align*}