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This activity is graded for completion so long as a relevant and genuine handwritten
attempt is made on each exercise. This activity is due in Gradescope by
11:59PM.
First Order Linear: A first order equation is linear if it can be transformed to
\(y^{\,\prime }+p(x)y=f(x)\). If \(f(x)=0\) it’s homogeneous and otherwise it’s nonhomogeneous.
Separable: A first order equation is separable if it can be transformed to
\(h(y)y^{\,\prime }=g(x)\).
[Note: Often \(t\) is used as the independent variable instead of \(x\).]
(a)
Categorize each of the following first order differential equation as either
(A) linear and not separable, (B) separable and not linear, (C) linear and
separable or (D) neither linear nor separable.
(i)
\(y^{\,\prime }=1-y^2\)
(ii)
\(y^{\,\prime }=ty-t\)
(iii)
\(y^{\,\prime }=1-x^2y\)
(iv)
\(y^{\,\prime }=\dfrac {t}{y}-y\)
(v)
\(y^{\,\prime }=t+y^3\)
(vi)
\(y^{\,\prime }=x\sin {(y)}\)
(vii)
\(y^{\,\prime }=(x-y)^2\)
(viii)
\(y^{\,\prime }=y\tan {(t)}\)
(ix)
\(y^{\,\prime }=2ye^{x-y^2}\)
(x)
\(y^{\,\prime }=\dfrac {x+y}{x}\)
(xi)
\(y^{\,\prime }=\dfrac {x+y}{y}\)
(xii)
\(y^{\,\prime }=x-y^{1/3}\)
(b)
The differential equation \(t^2y^{\,\prime }=1+y\) is both linear and separable.
(i)
Are there any constant solutions of the form \(y=C\)? If so, find them and
put the values of \(C\) in the box provided.
\(y=C\) for \(C=\)
(ii)
Solve this equation using the integrating factor method for a first
order linear nonhomogeneous equation. Show all steps! Can some or
all of the constant solutions (if any) from part \((a)\) be lumped together
with all these solutions? If not, include \(``\)or \(y=C_{1}\) or \(y=C_{2}\) or..." in the box
provided.
\(y=\)
(iii)
Now solve the same equation using the standard method for a
separable equation. Show all steps! Put the general solution found
in the box provided (remembering to address constant solutions if
needed). Did you get the same result as in \((b)\)?
\(y=\)
(iv)
An IVP of the form \(t^2y^{\,\prime }=1+y\), \(y(t_{0})=y_{0}\) will be guaranteed a unique solution (on some
open interval containing \(t_{0}\)) so long as \(t_{0}\neq 0\).
Explain why it is that with this knowledge the IVP \(t^2y^{\,\prime }=1+y\), \(y(1)=-1\) can be solved
without doing much work (such as the work done in part \((b)\) or \((c)\)). Would
have it been apparent if one had just done part \((b)\) or \((c)\)?
(c)
The differential equation \(xy^{\,\prime }=-y-y^{2}\) is not linear, but is separable.
(i)
Are there any constant solutions of the form \(y=C\)? If so, find them and
put the values of \(C\) in the box provided.
\(y=C\) for \(C=\)
(ii)
Solve this equation using the standard method for a separable
equation. Show all steps! Put the general solution found in the box
provided. Can some or all of the constant solutions (if any) from part
\((a)\) be lumped together with all these solutions? If not, include \(``\)or \(y=C_{1}\) or \(y=C_{2}\)
or..."in the box provided.
\(y=\)
(iii)
An IVP of the form \(xy^{\,\prime }=-y-y^2\), \(y(x_{0})=y_{0}\) will be guaranteed a unique solution (on some
open interval containing \(x_{0}\)) so long as \(x_{0}\neq 0\).
Explain why it is that with this knowledge the IVP \(xy^{\,\prime }=-y-y^2\), \(y(1)=0\) can be solved
without much work (such as the work done in part \((b)\)). Would this
solution have been apparent if one had just done part \((b)\)?
(d)
The differential \(y^{\,\prime }=2x+y\) is not separable, but is linear.
(i)
Are there any constant solutions of the form \(y=C\)? If so, find them and
put the values of \(C\) in the box provided.
\(y=C\) for \(C=\)
(ii)
Solve this equation using the integrating factor method for a first
order linear nonhomogeneous equation. Show all steps! Put the
general solution found in the box provided. Can some or all of the
constant solutions (if any) from part \((a)\) be lumped together with all
these solutions? If not, include \(``\)or \(y=C_{1}\) or \(y=C_{2}\) or..."in the box provided.
\(y=\)
(iii)
An IVP of the form \(y^{\,\prime }=2x+y\), \(y(x_{0})=y_{0}\) will ALWAYS be guaranteed a unique solution
(on any open interval containing \(x_{0}\)).
Find the solution to the IVP \(y^{\,\prime }=2x+y\), \(y(0)=0\). Put the solution in the box provided.
\(y=\)
(e)
Solve each IVP below, and for each report the open interval \(I\) on which the
solution is valid, but not so on any interval \(J\neq I\) containing \(I\).
Hints: the interval must contain the independent variable coordinate
of initial condition, the differential equation must be defined on the
interval AND the solution must be differentiable on the interval. Put
the solutions in the boxes provided. Look for ones that require no
work!
Applied Differential Equations