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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.
(a)
Consider a so-called Bernoulli equation: \(y^{\,\prime }+p(x)y=f(x)y^{n}\) where \(n\) is a real number not equal
to \(0\) nor \(1\).
(i)
For what values of \(n\) does a Bernoulli equation have a constant solution.
For those values, what constant solution(s) does a Bernoulli equation
have? Put the answer in the box provided.
(ii)
Find the general solution to the Bernoulli equation \(y^{\,\prime }+y=\dfrac {y^3}{e^{x}}\). Put the answer
in the box provided.
Hints: make note of any constant solution(s). You can use that \(y_1=e^{-x}\) is a solution to the complementary equation together with the
substitution \(u=\dfrac {y}{y_1}\) to transform the equation into a separable equation.
Alternatively, the substitution \(v=y^{-2}\) will transform he equation into a
linear equation..
\(y=\)
(iii)
Solve the IVP \(y^{\,\prime }+y=\dfrac {y^3}{e^{x}}\), \(y(0)=-1\). Put the answer in the box provided.
(iv)
Solve the IVP \(y^{\,\prime }+y=\dfrac {y^3}{e^{x}}\), \(y(0)=0\). Put the answer in the box provided.
(b)
A nonlinear homogeneous first order equation is a nonlinear differential
equation of the form \(y^{\,\prime }=f(y/x)\). By applying the substitution \(u=y/x\) we can transform this type
of equation to a separable differential equation..
(i)
Solve \(y^{\,\prime }=\dfrac {x^{2}-xy+y^{2}}{x^{2}}\). Show all your work. Put the answer in the box provided.
(ii)
Solve the IVP \(y^{\,\prime }=\dfrac {x^{2}-xy+y^{2}}{x^{2}}\), \(y(1)=0\). Show all your work. Put the answer in the box
provided.
(c)
Another kind of substitution:
(i)
Consider \(y^{\,\prime }=(2x-y+3)^{2}\). What transformation do you see that might be useful? Put
the answer in the box provided.
\(u=\)
(ii)
Transform the equation! Put the resulting equation in the box
provided.
(iii)
Solve for \(u\) and then \(y\). Put the solutions in the box provided.
\(u=\) \(y=\)
(d)
Now consider the first order differential equation \((y^{\,\prime }(t))^{2}=t+y\). Try to solve it for an
implicit general solution of the form \(F(t,y)=C\). Put your answer in the box
provided.
Hint: \(\displaystyle \int \dfrac {1}{1\pm \sqrt {u}}\;du\) can be evaluated with a substitution for the denominator.
Applied Differential Equations