Internal
problem
ID
[8313] Book
:
Own
collection
of
miscellaneous
problems Section
:
section
3.0 Problem
number
:
29 Date
solved
:
Sunday, November 10, 2024 at 03:38:12 AM CAS
classification
:
[[_homogeneous, `class D`]]
Where \(b\) is scalar and \(g\left ( x\right ) \) is function of \(x\) and \(n,m\) are integers. The
solution is given in Kamke page 20. Using the substitution \(y\left ( x\right ) =u\left ( x\right ) x\) then
The above ode is always separable. This is easily solved for \(u\) assuming the integration can be
resolved, and then the solution to the original ode becomes \(y=ux\). Comapring the given ode (A)
with the form (1) shows that
\begin{align*} g \left (x \right )&=2 x^{2}\\ b&=1\\ f \left (\frac {b x}{y}\right )&=\sin \left (\frac {y}{x}\right ) \end{align*}
Substituting the above in (2) results in the \(u(x)\) ode as
\begin{align*} u^{\prime }\left (x \right ) = 2 x \sin \left (u \left (x \right )\right )^{2} \end{align*}
Which is now solved as separable The ode \(u^{\prime }\left (x \right ) = 2 x \sin \left (u \left (x \right )\right )^{2}\) is separable as it can be written as
\begin{align*} u^{\prime }\left (x \right )&= 2 x \sin \left (u \left (x \right )\right )^{2}\\ &= f(x) g(u) \end{align*}
We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is
zero, since we had to divide by this above. Solving \(g(u)=0\) or \(\sin \left (u \right )^{2}=0\) for \(u \left (x \right )\) gives
\begin{align*} u \left (x \right )&=0 \end{align*}
Now we go over each such singular solution and check if it verifies the ode itself and
any initial conditions given. If it does not then the singular solution will not be
used.
Therefore the solutions found are
\begin{align*} -\cot \left (u \left (x \right )\right ) = x^{2}+c_1\\ u \left (x \right ) = 0 \end{align*}
We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is
zero, since we had to divide by this above. Solving \(g(u)=0\) or \(\sin \left (u \right )^{2}=0\) for \(u \left (x \right )\) gives
\begin{align*} u \left (x \right )&=0 \end{align*}
Now we go over each such singular solution and check if it verifies the ode itself and
any initial conditions given. If it does not then the singular solution will not be
used.
Therefore the solutions found are
\begin{align*} -\cot \left (u \left (x \right )\right ) = x^{2}+c_1\\ u \left (x \right ) = 0 \end{align*}
`Methodsfor first order ODEs:---Trying classification methods ---tryinga quadraturetrying1st order lineartryingBernoullitryingseparabletryinginverse lineartryinghomogeneous types:tryinghomogeneous D<-homogeneous successful`