Exercises

(1)
A first order differential equation is nonlinear homogeneous if it is a nonlinear differential equation of the form \(y^{\,\prime }=f(y/x)\) for some function \(f\). Exactly one of the following equations is NOT nonlinear homogeneous. Which one?
(a)
\(\displaystyle y^{\,\prime }=1+\dfrac {x}{y}\)
(b)
\(\displaystyle y^{\,\prime }=\dfrac {2x}{x+3y}\)
(c)
\(\displaystyle y^{\,\prime }=\dfrac {2x^2}{x^2+y^2}\)
(d)
\(\displaystyle y^{\,\prime }=1+\dfrac {y}{x}\)
(e)
\(\displaystyle y^{\,\prime }=e^{\dfrac {x+y}{x}}\)
(2)
Suppose \(y(x)\), for \(x>0\), satisfies \(y^{\,\prime }=\dfrac {y}{x-y}\), \(y(1)=0\). Find \(y(2026)\) to the nearest hundredth.
(3)
Given the nonlinear homogeneous de \(\displaystyle y^{\,\prime }=\dfrac {x^2+y^2}{xy}\) determine the transformed separable equation that is the result of applying the substitution \(u=\frac {y}{x}\).
(a)
\(\displaystyle u u'=\frac {1}{x}\)
(b)
\(\displaystyle \frac {1}{u} u'=\frac {1}{x}\)
(c)
\(\displaystyle \frac {1}{u^2} u'=-\frac {1}{x}\)
(d)
\(\displaystyle \frac {u}{1+u^2} u'=\frac {1}{x}\)
(e)
\(\displaystyle \frac {u^2}{1-u^3} u'=-\frac {1}{x}\)
(4)
Suppose \(y(x)\) satisfies the nonlinear homogeneous IVP \(\displaystyle y^{\,\prime }=\dfrac {x^2+y^2}{xy}\), \(y(1)=1\). Find \(y(2026)\) to the nearest tenth.
(5)
Suppose \(y(x)\) satisfies the nonlinear homogeneous IVP \(\displaystyle y^{\,\prime }=\dfrac {x^2+3xy+y^2}{x^2}\), \(y(1)=a\), where \(a\) is a constant. Given that \(y(e)=e\) find \(a\) to the nearest hundredth.