Internal problem ID [2315]
Internal file name [OUTPUT/2315_Tuesday_February_27_2024_08_25_57_AM_60061494/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 37, page 171
Problem number: 2.
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]
\[ \boxed {x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }=-1} \] The ode \begin {align*} x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime } = -1 \end {align*}
is factored to \begin {align*} \left (y^{\prime } x +1\right ) \left (y y^{\prime }+1\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } x +1 = 0\tag {1} \\ y y^{\prime }+1 = 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*} \int -y d y &= x +c_{2}\\ -\frac {y^{2}}{2}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {-2 x -2 c_{2}}\\ y_2&=-\sqrt {-2 x -2 c_{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-2 x -2 c_{2}} \\ \tag{2} y &= -\sqrt {-2 x -2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-2 x -2 c_{2}} \] Verified OK.
\[ y = -\sqrt {-2 x -2 c_{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-2 x -2 c_{2}} \\ \tag{2} y &= -\sqrt {-2 x -2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-2 x -2 c_{2}} \] Verified OK.
\[ y = -\sqrt {-2 x -2 c_{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {1}{x}, y^{\prime }=-\frac {1}{y}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {1}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\ln \left (x \right )+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {1}{y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y y^{\prime }=-1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y y^{\prime }d x =\int \left (-1\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-2 x +2 c_{1}}, y=-\sqrt {-2 x +2 c_{1}}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\ln \left (x \right )+c_{1} , \left \{y=\sqrt {-2 x +2 c_{1}}, y=-\sqrt {-2 x +2 c_{1}}\right \}\right \} \end {array} \]
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 trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
dsolve(x*y(x)*diff(y(x),x)^2+(x+y(x))*diff(y(x),x)+1=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\ln \left (x \right )+c_{1} \\ y \left (x \right ) &= \sqrt {-2 x +c_{1}} \\ y \left (x \right ) &= -\sqrt {-2 x +c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 53
DSolve[x*y[x]*y'[x]^2+(x+y[x])*y'[x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {-x+c_1} \\ y(x)\to \sqrt {2} \sqrt {-x+c_1} \\ y(x)\to -\log (x)+c_1 \\ \end{align*}