Internal problem ID [8779]
Internal file name [OUTPUT/7714_Sunday_June_05_2022_11_45_59_PM_54417012/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 445.
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]
\[ \boxed {x^{2} {y^{\prime }}^{2}+\left (y^{3} a \,x^{2}+b \right ) y^{\prime }+y^{3} a b=0} \] The ode \begin {align*} x^{2} {y^{\prime }}^{2}+\left (y^{3} a \,x^{2}+b \right ) y^{\prime }+y^{3} a b = 0 \end {align*}
is factored to \begin {align*} \left (x^{2} y^{\prime }+b \right ) \left (a y^{3}+y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} x^{2} y^{\prime }+b = 0\tag {1} \\ a y^{3}+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 {b}{x^{2}}\,\mathop {\mathrm {d}x}}\\ &= \frac {b}{x}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {b}{x}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {b}{x}+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {b}{x}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {b}{x}+c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} \int -\frac {1}{a \,y^{3}}d y &= x +c_{2}\\ \frac {1}{2 y^{2} a}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{\sqrt {2 a c_{2} +2 x a}}\\ y_2&=-\frac {1}{\sqrt {2 a c_{2} +2 x a}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{\sqrt {2 a c_{2} +2 x a}} \\ \tag{2} y &= -\frac {1}{\sqrt {2 a c_{2} +2 x a}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{\sqrt {2 a c_{2} +2 x a}} \] Verified OK.
\[ y = -\frac {1}{\sqrt {2 a c_{2} +2 x a}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{\sqrt {2 a c_{2} +2 x a}} \\ \tag{2} y &= -\frac {1}{\sqrt {2 a c_{2} +2 x a}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{\sqrt {2 a c_{2} +2 x a}} \] Verified OK.
\[ y = -\frac {1}{\sqrt {2 a c_{2} +2 x a}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} {y^{\prime }}^{2}+\left (y^{3} a \,x^{2}+b \right ) y^{\prime }+y^{3} a b =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {b}{x^{2}}, y^{\prime }=-a y^{3}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {b}{x^{2}} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {b}{x^{2}}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {b}{x}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {c_{1} x +b}{x} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-a y^{3} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{3}}=-a \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{3}}d x =\int -a d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{2 y^{2}}=-x a +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {1}{\sqrt {2 x a -2 c_{1}}}, y=-\frac {1}{\sqrt {2 x a -2 c_{1}}}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {c_{1} x +b}{x}, \left \{y=\frac {1}{\sqrt {2 x a -2 c_{1}}}, y=-\frac {1}{\sqrt {2 x a -2 c_{1}}}\right \}\right \} \end {array} \]
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 <- quadrature successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 35
dsolve(x^2*diff(y(x),x)^2+(a*x^2*y(x)^3+b)*diff(y(x),x)+a*b*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {2 a x +c_{1}}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {2 a x +c_{1}}} \\ y \left (x \right ) &= \frac {b}{x}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 49
DSolve[a*b*y[x]^3 + (b + a*x^2*y[x]^3)*y'[x] + x^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{\sqrt {2 a x-2 c_1}} \\ y(x)\to \frac {1}{\sqrt {2 a x-2 c_1}} \\ y(x)\to \frac {b}{x}+c_1 \\ \end{align*}