Internal problem ID [6161]
Internal file name [OUTPUT/5409_Sunday_June_05_2022_03_36_31_PM_35181252/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.4 First Order Linear Equations.
Page 15
Problem number: 1(e).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {2 y-x y^{\prime }=x^{3}} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=-\frac {2}{x}\\ q(x) &=-x^{2} \end {align*}
Hence the ode is \begin {align*} y^{\prime }-\frac {2 y}{x} = -x^{2} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int -\frac {2}{x}d x} \\ &= \frac {1}{x^{2}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (-x^{2}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\frac {y}{x^{2}}\right ) &= \left (\frac {1}{x^{2}}\right ) \left (-x^{2}\right )\\ \mathrm {d} \left (\frac {y}{x^{2}}\right ) &= -1\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \frac {y}{x^{2}} &= \int {-1\,\mathrm {d} x}\\ \frac {y}{x^{2}} &= -x + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\frac {1}{x^{2}}\) results in \begin {align*} y &= c_{1} x^{2}-x^{3} \end {align*}
which simplifies to \begin {align*} y &= x^{2} \left (c_{1} -x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x^{2} \left (c_{1} -x \right ) \\ \end{align*}
Verification of solutions
\[ y = x^{2} \left (c_{1} -x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y-x y^{\prime }=x^{3} \\ \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 }=-\frac {-2 y+x^{3}}{x} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 y}{x}-x^{2} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {2 y}{x}=-x^{2} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }-\frac {2 y}{x}\right )=-\mu \left (x \right ) x^{2} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (y \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }-\frac {2 y}{x}\right )=y^{\prime } \mu \left (x \right )+y \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=-\frac {2 \mu \left (x \right )}{x} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\frac {1}{x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (y \mu \left (x \right )\right )\right )d x =\int -\mu \left (x \right ) x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int -\mu \left (x \right ) x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int -\mu \left (x \right ) x^{2}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\frac {1}{x^{2}} \\ {} & {} & y=x^{2} \left (\int \left (-1\right )d x +c_{1} \right ) \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=x^{2} \left (c_{1} -x \right ) \end {array} \]
Maple trace
`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: 13
dsolve(2*y(x)-x^3=x*diff(y(x),x),y(x), singsol=all)
\[ y \left (x \right ) = \left (-x +c_{1} \right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 15
DSolve[2*y[x]-x^3==x*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^2 (-x+c_1) \]