Internal problem ID [5089]
Internal file name [OUTPUT/4582_Sunday_June_05_2022_03_01_18_PM_46227135/index.tex]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page
1068
Problem number: 3.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\left (y+1\right )^{2} y^{\prime }=-x^{3}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {x^{3}}{\left (y +1\right )^{2}} \end {align*}
Where \(f(x)=-x^{3}\) and \(g(y)=\frac {1}{\left (y +1\right )^{2}}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{\left (y +1\right )^{2}}} \,dy &= -x^{3} \,d x \\ \int { \frac {1}{\frac {1}{\left (y +1\right )^{2}}} \,dy} &= \int {-x^{3} \,d x} \\ \frac {\left (y +1\right )^{3}}{3}&=-\frac {x^{4}}{4}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{2}-1 \\ y &= -\frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \\ y &= -\frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{2}-1 \\ \tag{2} y &= -\frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \\ \tag{3} y &= -\frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{2}-1 \] Verified OK.
\[ y = -\frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \] Verified OK.
\[ y = -\frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y+1\right )^{2} y^{\prime }=-x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (y+1\right )^{2} y^{\prime }d x =\int -x^{3}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\left (y+1\right )^{3}}{3}=-\frac {x^{4}}{4}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (-6 x^{4}+24 c_{1} \right )^{\frac {1}{3}}}{2}-1 \end {array} \]
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: 0.0 (sec). Leaf size: 89
dsolve(x^3+(y(x)+1)^2*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-6 x^{4}-24 c_{1} \right )^{\frac {1}{3}}}{2}-1 \\ y \left (x \right ) &= -\frac {\left (-6 x^{4}-24 c_{1} \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (-6 x^{4}-24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \\ y \left (x \right ) &= -\frac {\left (-6 x^{4}-24 c_{1} \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (-6 x^{4}-24 c_{1} \right )^{\frac {1}{3}}}{4}-1 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.483 (sec). Leaf size: 110
DSolve[x^3+(y[x]+1)^2*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -1+\frac {\sqrt [3]{-3 x^4+4+12 c_1}}{2^{2/3}} \\ y(x)\to -1+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-3 x^4+4+12 c_1}}{2\ 2^{2/3}} \\ y(x)\to -1-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-3 x^4+4+12 c_1}}{2\ 2^{2/3}} \\ \end{align*}