Internal problem ID [4764]
Internal file name [OUTPUT/4257_Sunday_June_05_2022_12_48_52_PM_7362242/index.tex]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 3. Linear First-Order Equations. page
403
Problem number: 4.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {2 x y^{\prime }+y=2 x^{\frac {5}{2}}} \]
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 {1}{2 x}\\ q(x) &=x^{\frac {3}{2}} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {y}{2 x} = x^{\frac {3}{2}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {1}{2 x}d x} \\ &= \sqrt {x} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (x^{\frac {3}{2}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\sqrt {x}\, y\right ) &= \left (\sqrt {x}\right ) \left (x^{\frac {3}{2}}\right )\\ \mathrm {d} \left (\sqrt {x}\, y\right ) &= x^{2}\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \sqrt {x}\, y &= \int {x^{2}\,\mathrm {d} x}\\ \sqrt {x}\, y &= \frac {x^{3}}{3} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\sqrt {x}\) results in \begin {align*} y &= \frac {x^{\frac {5}{2}}}{3}+\frac {c_{1}}{\sqrt {x}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{\frac {5}{2}}}{3}+\frac {c_{1}}{\sqrt {x}} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{\frac {5}{2}}}{3}+\frac {c_{1}}{\sqrt {x}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x y^{\prime }+y=2 x^{\frac {5}{2}} \\ \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 {-y+2 x^{\frac {5}{2}}}{2 x} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y}{2 x}+x^{\frac {3}{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 {y}{2 x}=x^{\frac {3}{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 {y}{2 x}\right )=\mu \left (x \right ) x^{\frac {3}{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 {y}{2 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 {\mu \left (x \right )}{2 x} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\sqrt {x} \\ \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^{\frac {3}{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^{\frac {3}{2}}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right ) x^{\frac {3}{2}}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\sqrt {x} \\ {} & {} & y=\frac {\int x^{2}d x +c_{1}}{\sqrt {x}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\frac {x^{3}}{3}+c_{1}}{\sqrt {x}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\frac {x^{3}+3 c_{1}}{3 \sqrt {x}} \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: 16
dsolve(2*x*diff(y(x),x)+y(x)=2*x^(5/2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{3}+3 c_{1}}{3 \sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 22
DSolve[2*x*y'[x]+y[x]==2*x^(5/2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^3+3 c_1}{3 \sqrt {x}} \]