Chapter 4

Suppose \(y_{1}, y_{2}, \ldots, y_{k}\) are \(k\) linearly independent solutions on \((-\infty, \infty)\) of a homogeneous linear \(n\) th-order differential equation with constant coefficients. By Theorem 4.1 .2 it follows that \(y_{k+1}=0\) is also a solution of the differential equation. Is the set of solutions \(y_{1}, y_{2}, \ldots, y_{k}, y_{k+1}\) linearly dependent or linearly independent on \((-\infty, \infty) ?\) Discuss.

Solve the given initial-value problem in which the input function \(g(x)\) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that \(y\) and \(y^{\prime}\) are continuous at \(x=\pi / 2\) (Problem 41 ) and at \(x=\pi\) (Problem 42 ). $$\begin{aligned}&y^{\prime \prime}-2 y^{\prime}+10 y=g(x), \quad y(0)=0, y^{\prime}(0)=0, \quad \text { where }\\\&g(x)=\left\\{\begin{array}{ll}20, & 0 \leq x \leq \pi \\\0, & x>\pi\end{array}\right.\end{aligned}$$

Use the substitution \(t=x-x_{0}\) to solve the given differential equation. $$(x-4)^{2} y^{\prime \prime}-5(x-4) y^{\prime}+9 y=0$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-2 y^{\prime}+y=x^{3}+4 x$$

Suppose that \(y_{1}, y_{2}, \ldots, y_{k}\) are \(k\) nontrivial solutions of a homogeneous linear \(n\) th-order differential equation with constant coefficients and that \(k=n+1 .\) Is the set of solutions \(y_{1}, y_{2}, \ldots, y_{k}\) linearly dependent or linearly independent on \((-\infty, \infty) ?\) Discuss.

Consider the differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=e^{k x}\) where \(a, b, c,\) and \(k\) are constants. The auxiliary equation of the associated homogeneous equation is \(a m^{2}+b m+c=0\). (a) If \(k\) is not a root of the auxiliary equation, show that we can find a particular solution of the form \(y_{p}=A e^{k x},\) where \(A=1 /\left(a k^{2}+b k+c\right)\) (b) If \(k\) is a root of the auxiliary equation of multiplicity one, show that we can find a particular solution of the form \(y_{p}=A x e^{k x},\) where \(A=1 /(2 a k+b) .\) Explain how we know that \(k \neq-b /(2 a)\) (c) If \(k\) is a root of the auxiliary equation of multiplicity two, show that we can find a particular solution of the form \(y=A x^{2} e^{k x},\) where \(A=1 /(2 a)\)

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-y^{\prime}-12 y=e^{4 x}$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+2 y^{\prime}+2 y=5 e^{6 x}$$

The initial-conditions \(y(0)=y_{0}, y^{\prime}(0)=y_{1}\) apply to each of the following differential equations: $$\begin{aligned}&x^{2} y^{\prime \prime}=0\\\&\begin{array}{l}x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=0 \\\x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0\end{array}\end{aligned}$$ For what values of \(y_{0}\) and \(y_{1}\) does each initial-value problem have a solution?

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-2 y^{\prime}-3 y=4 e^{x}-9$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+6 y^{\prime}+8 y=3 e^{-2 x}+2 x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+25 y=6 \sin x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+4 y=4 \cos x+3 \sin x-8$$

Find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra. $$\begin{aligned}y^{\prime \prime}-4 y^{\prime}+8 y=&\left(2 x^{2}-3 x\right) e^{2 x} \cos 2 x \\\&+\left(10 x^{2}-x-1\right) e^{2 x} \sin 2 x\end{aligned}$$

Solve the given differential equation by using a CAS to find the (approximate) roots of the auxiliary equation. $$2 x^{3} y^{\prime \prime \prime}-10.98 x^{2} y^{\prime \prime}+8.5 x y^{\prime}+1.3 y=0$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} e^{x}+c_{2} e^{5 x}$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+6 y^{\prime}+9 y=-x e^{4 x}$$

Find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra. $$y^{(4)}+2 y^{\prime \prime}+y=2 \cos x-3 x \sin x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+3 y^{\prime}-10 y=x\left(e^{x}+1\right)$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} e^{-4 x}+c_{2} e^{-3 x}$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1}+c_{2} e^{2 x}$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-y=x^{2} e^{x}+5$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+2 y^{\prime}+y=x^{2} e^{-x}$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} e^{10 x}+c_{2} x e^{10 x}$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} \cos 3 x+c_{2} \sin 3 x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-2 y^{\prime}+5 y=e^{x} \sin x$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} \cosh 7 x+c_{2} \sinh 7 x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+y^{\prime}+\frac{1}{4} y=e^{x}(\sin 3 x-\cos 3 x)$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} e^{-x} \cos x+c_{2} e^{-x} \sin x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+25 y=20 \sin 5 x$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1}+c_{2} e^{2 x} \cos 5 x+c_{3} e^{2 x} \sin 5 x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+y=4 \cos x-\sin x$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1}+c_{2} x+c_{3} e^{8 x}$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+y^{\prime}+y=x \sin x$$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$y=c_{1} \cos x+c_{2} \sin x+c_{3} \cos 2 x+c_{4} \sin 2 x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+4 y=\cos ^{2} x$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime \prime}+8 y^{\prime \prime}=-6 x^{2}+9 x+2$$

Find the general solution of \(2 y^{\prime \prime \prime}+7 y^{\prime \prime}+4 y^{\prime}-4 y=0\) if \(m_{1}=\frac{1}{2}\) is one root of its auxiliary equation.

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime \prime}-y^{\prime \prime}+y^{\prime}-y=x e^{x}-e^{-x}+7$$

Find the general solution of \(y^{\prime \prime \prime}+6 y^{\prime \prime}+y^{\prime}-34 y=0\) if it is known that \(y_{1}=e^{-4 x} \cos x\) is one solution.

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=e^{x}-x+16$$

To solve \(y^{(4)}+y=0,\) we must find the roots of \(m^{4}+1=0\) This is a trivial problem using a CAS but can also be done by hand working with complex numbers. Observe that \(m^{4}+1=\left(m^{2}+1\right)^{2}-2 m^{2} .\) How does this help? Solve the differential equation.

Solve the given differential equation by undetermined coefficients. $$2 y^{\prime \prime \prime}-3 y^{\prime \prime}-3 y^{\prime}+2 y=\left(e^{x}+e^{-x}\right)^{2}$$

Verify that \(y=\sinh x-2 \cos (x+\pi / 6)\) is a particular solution of \(y^{(4)}-y=0 .\) Reconcile this particular solution with the general solution of the DE.

Solve the given differential equation by undetermined coefficients. $$y^{(4)}-2 y^{\prime \prime \prime}+y^{\prime \prime}=e^{x}+1$$

Consider the boundary-value problem \(y^{\prime \prime}+\lambda y=0, y(0)=0\) \(y(\pi / 2)=0 .\) Discuss: Is it possible to determine real values of \(\lambda\) so that the problem possesses (a) trivial solutions? (b) nontrivial solutions? the general solution, simplify the output and, if necessary, write the solution in terms of real functions.

Solve the given differential equation by undetermined coefficients. $$y^{(4)}-4 y^{\prime \prime}=5 x^{2}-e^{2 x}$$

$$y^{\prime \prime \prime}-6 y^{\prime \prime}+2 y^{\prime}+y=0$$

Solve the given initial-value problem. $$y^{\prime \prime}-64 y=16, \quad y(0)=1, y^{\prime}(0)=0$$

$$6.11 y^{\prime \prime \prime}+8.59 y^{\prime \prime}+7.93 y^{\prime}+0.778 y=0$$