Exercises

(1)
Which first order differential equation(s) are NOT separable? Select all that apply.
(a)
\(y^{\,\prime }=1-y^2\)
(b)
\(y^{\,\prime }=\dfrac {x}{y}+x\)
(c)
\(y^{\,\prime }=\dfrac {y}{x}+x\)
(d)
\(y^{\,\prime }=\tan ^{-1}(xy)\)
(e)
\(y^{\,\prime }=e^{x^2-y}\)
(f)
\(y^{\,\prime }=y^2+xy^2\)
(g)
\(y^{\,\prime }=y^3+xy^2\)
(h)
\(y^{\,\prime }=\ln \left (x^y\right )\)
(2)
There is a solution of the form \(y=\dfrac {1}{C-x}\) for the first order IVP \(y^{\,\prime }=y^2\), \(y(1)=0.5\). Find the constant \(C\).
(3)
The differential equation \(y^{\,\prime }=2x(y^2-4)\ln (y)\) is separable. How many constant solutions of the form \(y=C\) does this differential equation have?
(4)
Find the width of the widest interval on which there exists a solution to the IVP \(y^{\,\prime }=-\dfrac {y^2}{x}\), \(y(e^{-2})=-1\). Round to the nearest hundredth.
(5)
Suppose \(y(x)\) satisfies the first order IVP \((x^2+a^2)\,y^{\,\prime }=y^3-3b y^2+3b^2y-b^3\), \(y(0)=b\) where \(a,b\) are non-zero constants. Which is \(y(2026)\)?
(a)
\(a\)
(b)
\(b\)
(c)
\(0\)
(d)
\(a+b\)
(e)
\(a-b\)
(f)
\(ab\)
(g)
\(a/b\)
(h)
None of these