Internal problem ID [4149]
Internal file name [OUTPUT/3642_Sunday_June_05_2022_09_56_22_AM_34574844/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 913.
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]
\[ \boxed {x^{2} {y^{\prime }}^{2}+\left (a +b \,x^{2} y^{3}\right ) y^{\prime }+a b y^{3}=0} \] The ode \begin {align*} x^{2} {y^{\prime }}^{2}+\left (a +b \,x^{2} y^{3}\right ) y^{\prime }+a b y^{3} = 0 \end {align*}
is factored to \begin {align*} \left (y^{3} b +y^{\prime }\right ) \left (y^{\prime } x^{2}+a \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{3} b +y^{\prime } = 0\tag {1} \\ y^{\prime } x^{2}+a = 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^{3} b}d y &= x +c_{1}\\ \frac {1}{2 y^{2} b}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{\sqrt {2 b c_{1} +2 b x}}\\ y_2&=-\frac {1}{\sqrt {2 b c_{1} +2 b x}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{\sqrt {2 b c_{1} +2 b x}} \\ \tag{2} y &= -\frac {1}{\sqrt {2 b c_{1} +2 b x}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{\sqrt {2 b c_{1} +2 b x}} \] Verified OK.
\[ y = -\frac {1}{\sqrt {2 b c_{1} +2 b x}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{\sqrt {2 b c_{1} +2 b x}} \\ \tag{2} y &= -\frac {1}{\sqrt {2 b c_{1} +2 b x}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{\sqrt {2 b c_{1} +2 b x}} \] Verified OK.
\[ y = -\frac {1}{\sqrt {2 b c_{1} +2 b x}} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} y &= \int { -\frac {a}{x^{2}}\,\mathop {\mathrm {d}x}}\\ &= \frac {a}{x}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {a}{x}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {a}{x}+c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {a}{x}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {a}{x}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} {y^{\prime }}^{2}+\left (a +b \,x^{2} y^{3}\right ) y^{\prime }+a b y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {a}{x^{2}}, y^{\prime }=-y^{3} b \right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {a}{x^{2}} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {a}{x^{2}}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {a}{x}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {c_{1} x +a}{x} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-y^{3} b \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{3}}=-b \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{3}}d x =\int -b d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{2 y^{2}}=-b x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {1}{\sqrt {2 b x -2 c_{1}}}, y=-\frac {1}{\sqrt {2 b x -2 c_{1}}}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {c_{1} x +a}{x}, \left \{y=\frac {1}{\sqrt {2 b x -2 c_{1}}}, y=-\frac {1}{\sqrt {2 b x -2 c_{1}}}\right \}\right \} \end {array} \]
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: 35
dsolve(x^2*diff(y(x),x)^2+(a+b*x^2*y(x)^3)*diff(y(x),x)+a*b*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {a}{x}+c_{1} \\ y \left (x \right ) &= \frac {1}{\sqrt {2 b x +c_{1}}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {2 b x +c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 49
DSolve[x^2 (y'[x])^2+(a+b x^2 y[x]^3)y'[x]+a b y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{\sqrt {2 b x-2 c_1}} \\ y(x)\to \frac {1}{\sqrt {2 b x-2 c_1}} \\ y(x)\to \frac {a}{x}+c_1 \\ \end{align*}