Internal problem ID [6772]
Internal file name [OUTPUT/6019_Monday_July_25_2022_01_59_42_AM_10893776/index.tex]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES
Page 309
Problem number: 6.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "linear", "quadrature", "separable", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{y^{\prime }}^{2}-\left (y x^{2}+3\right ) y^{\prime }+3 y x^{2}=0} \] The ode \begin {align*} {y^{\prime }}^{2}-\left (y x^{2}+3\right ) y^{\prime }+3 y x^{2} = 0 \end {align*}
is factored to \begin {align*} \left (y^{\prime }-3\right ) \left (y x^{2}-y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }-3 = 0\tag {1} \\ y x^{2}-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 { 3\,\mathop {\mathrm {d}x}}\\ &= 3 x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 3 x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = 3 x +c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 3 x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = 3 x +c_{1} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= y \,x^{2} \end {align*}
Where \(f(x)=x^{2}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= x^{2} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {x^{2} \,d x}\\ \ln \left (y \right )&=\frac {x^{3}}{3}+c_{2}\\ y&={\mathrm e}^{\frac {x^{3}}{3}+c_{2}}\\ &=c_{2} {\mathrm e}^{\frac {x^{3}}{3}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{\frac {x^{3}}{3}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{\frac {x^{3}}{3}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{\frac {x^{3}}{3}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{\frac {x^{3}}{3}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-\left (y x^{2}+3\right ) y^{\prime }+3 y x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=3, y^{\prime }=y x^{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=3 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 3d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=3 x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=3 x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=y x^{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=x^{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int x^{2}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\frac {x^{3}}{3}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{\frac {x^{3}}{3}+c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=3 x +c_{1} , y={\mathrm e}^{\frac {x^{3}}{3}+c_{1}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
dsolve(diff(y(x),x)^2-(x^2*y(x)+3)*diff(y(x),x)+3*x^2*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} {\mathrm e}^{\frac {x^{3}}{3}} \\ y \left (x \right ) &= 3 x +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 27
DSolve[(y'[x])^2-(x^2*y[x]+3)*y'[x]+3*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{\frac {x^3}{3}} \\ y(x)\to 3 x+c_1 \\ \end{align*}