Internal problem ID [1148]
Internal file name [OUTPUT/1149_Sunday_June_05_2022_02_03_34_AM_91257170/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page
253
Problem number: 38 part (g).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {36 y^{\prime }+36 y^{2}-12 y=-1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}+\frac {1}{3} y -\frac {1}{36}}d y &= x +c_{1}\\ \frac {1}{y -\frac {1}{6}}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {c_{1} +x +6}{6 x +6 c_{1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} +x +6}{6 x +6 c_{1}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} +x +6}{6 x +6 c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 36 y^{\prime }+36 y^{2}-12 y=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}+\frac {y}{3}-\frac {1}{36} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{2}+\frac {y}{3}-\frac {1}{36}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{2}+\frac {y}{3}-\frac {1}{36}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {6}{6 y-1}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {c_{1} +x +6}{6 \left (x +c_{1} \right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 18
dsolve(36*diff(y(x),x)+36*y(x)^2-12*y(x)+1=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} +x +6}{6 x +6 c_{1}} \]
✓ Solution by Mathematica
Time used: 0.119 (sec). Leaf size: 30
DSolve[36*y'[x]+36*y[x]^2-12*y[x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x+6-36 c_1}{6 x-216 c_1} \\ y(x)\to \frac {1}{6} \\ \end{align*}