Internal problem ID [5714]
Internal file name [OUTPUT/4962_Sunday_June_05_2022_03_15_19_PM_32792222/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems.
page 7
Problem number: 1.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {x^{2}}{y}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x^{2}}{y} \end {align*}
Where \(f(x)=x^{2}\) and \(g(y)=\frac {1}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y}} \,dy &= x^{2} \,d x \\ \int { \frac {1}{\frac {1}{y}} \,dy} &= \int {x^{2} \,d x} \\ \frac {y^{2}}{2}&=\frac {x^{3}}{3}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \\ y &= -\frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \\ \tag{2} y &= -\frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \] Verified OK.
\[ y = -\frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {x^{2}}{y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y y^{\prime }=x^{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y y^{\prime }d x =\int x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=\frac {x^{3}}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {\sqrt {6 x^{3}+18 c_{1}}}{3}, y=\frac {\sqrt {6 x^{3}+18 c_{1}}}{3}\right \} \end {array} \]
Maple trace
`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: 33
dsolve(diff(y(x),x)=x^2/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6 x^{3}+9 c_{1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6 x^{3}+9 c_{1}}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.084 (sec). Leaf size: 50
DSolve[y'[x]==x^2/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {\frac {2}{3}} \sqrt {x^3+3 c_1} \\ y(x)\to \sqrt {\frac {2}{3}} \sqrt {x^3+3 c_1} \\ \end{align*}