Internal problem ID [8354]
Internal file name [OUTPUT/7287_Sunday_June_05_2022_05_42_39_PM_90737590/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 17.
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]
\[ \boxed {y^{\prime }-y^{2}-3 y=-4} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}+3 y -4}d y &= x +c_{1}\\ -\frac {\ln \left (y +4\right )}{5}+\frac {\ln \left (y -1\right )}{5}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {4 \,{\mathrm e}^{5 x +5 c_{1}}+1}{-1+{\mathrm e}^{5 x +5 c_{1}}}\\ &=-\frac {4 \,{\mathrm e}^{5 x} c_{1}^{5}+1}{-1+{\mathrm e}^{5 x} c_{1}^{5}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {4 \,{\mathrm e}^{5 x} c_{1}^{5}+1}{-1+{\mathrm e}^{5 x} c_{1}^{5}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {4 \,{\mathrm e}^{5 x} c_{1}^{5}+1}{-1+{\mathrm e}^{5 x} c_{1}^{5}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}-3 y=-4 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+3 y-4 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}+3 y-4}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}+3 y-4}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y-1\right )}{5}-\frac {\ln \left (y+4\right )}{5}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {4 \,{\mathrm e}^{5 x +5 c_{1}}+1}{-1+{\mathrm e}^{5 x +5 c_{1}}} \end {array} \]
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: 0.016 (sec). Leaf size: 24
dsolve(diff(y(x),x) - y(x)^2 -3*y(x) + 4=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-4 c_{1} {\mathrm e}^{5 x}-1}{-1+c_{1} {\mathrm e}^{5 x}} \]
✓ Solution by Mathematica
Time used: 0.49 (sec). Leaf size: 40
DSolve[y'[x] - y[x]^2 -3*y[x] + 4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-1-4 e^{5 (x+c_1)}}{-1+e^{5 (x+c_1)}} \\ y(x)\to -4 \\ y(x)\to 1 \\ \end{align*}