Internal problem ID [6866]
Internal file name [OUTPUT/6109_Friday_August_05_2022_02_19_54_AM_48557998/index.tex]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 2.
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]
\[ \boxed {6 x {y^{\prime }}^{2}-\left (3 x +2 y\right ) y^{\prime }+y=0} \] The ode \begin {align*} 6 x {y^{\prime }}^{2}-\left (3 x +2 y\right ) y^{\prime }+y = 0 \end {align*}
is factored to \begin {align*} \left (2 y^{\prime }-1\right ) \left (-3 y^{\prime } x +y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} 2 y^{\prime }-1 = 0\tag {1} \\ -3 y^{\prime } x +y = 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}{2}}\,\mathop {\mathrm {d}x}}\\ &= \frac {x}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x}{2}+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x}{2}+c_{1} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y}{3 x} \end {align*}
Where \(f(x)=\frac {1}{3 x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= \frac {1}{3 x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {\frac {1}{3 x} \,d x}\\ \ln \left (y \right )&=\frac {\ln \left (x \right )}{3}+c_{2}\\ y&={\mathrm e}^{\frac {\ln \left (x \right )}{3}+c_{2}}\\ &=c_{2} x^{\frac {1}{3}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x^{\frac {1}{3}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} x^{\frac {1}{3}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x^{\frac {1}{3}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} x^{\frac {1}{3}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 x {y^{\prime }}^{2}-\left (3 x +2 y\right ) y^{\prime }+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {1}{2}, y^{\prime }=\frac {y}{3 x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {1}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{2}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x}{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {y}{3 x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=\frac {1}{3 x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \frac {1}{3 x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\frac {\ln \left (x \right )}{3}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\left (\left ({\mathrm e}^{-3 c_{1}}\right )^{2} x \right )}^{\frac {1}{3}}}{{\mathrm e}^{-3 c_{1}}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {{\left (\left ({\mathrm e}^{-3 c_{1}}\right )^{2} x \right )}^{\frac {1}{3}}}{{\mathrm e}^{-3 c_{1}}}, y=\frac {x}{2}+c_{1} \right \} \end {array} \]
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: 17
dsolve(6*x*diff(y(x),x)^2-(3*x+2*y(x))*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} x^{\frac {1}{3}} \\ y \left (x \right ) &= \frac {x}{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 30
DSolve[6*x*(y'[x])^2-(3*x+2*y[x])*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \sqrt [3]{x} \\ y(x)\to \frac {x}{2}+c_1 \\ y(x)\to 0 \\ \end{align*}