Internal problem ID [5079]
Internal file name [OUTPUT/4572_Sunday_June_05_2022_03_01_06_PM_7648630/index.tex]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Test excercise 24. page 1067
Problem number: 5.
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]
\[ \boxed {x^{2} y^{\prime }=x^{3} \sin \left (3 x \right )+4} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {x^{3} \sin \left (3 x \right )+4}{x^{2}}\,\mathop {\mathrm {d}x}}\\ &= \frac {\sin \left (3 x \right )}{9}-\frac {x \cos \left (3 x \right )}{3}-\frac {4}{x}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sin \left (3 x \right )}{9}-\frac {x \cos \left (3 x \right )}{3}-\frac {4}{x}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sin \left (3 x \right )}{9}-\frac {x \cos \left (3 x \right )}{3}-\frac {4}{x}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }=x^{3} \sin \left (3 x \right )+4 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{3} \sin \left (3 x \right )+4}{x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {x^{3} \sin \left (3 x \right )+4}{x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\sin \left (3 x \right )}{9}-\frac {x \cos \left (3 x \right )}{3}-\frac {4}{x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {-3 x^{2} \cos \left (3 x \right )+x \sin \left (3 x \right )+9 c_{1} x -36}{9 x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 24
dsolve(x^2*diff(y(x),x)=x^3*sin(3*x)+4,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sin \left (3 x \right )}{9}-\frac {x \cos \left (3 x \right )}{3}-\frac {4}{x}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 30
DSolve[x^2*y'[x]==x^3*Sin[3*x]+4,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {4}{x}+\frac {1}{9} \sin (3 x)-\frac {1}{3} x \cos (3 x)+c_1 \]