Homework #4

1. (Rabenstein 1.4 #6 & #8 page 28 both 2^nd and 3^rd editions) Solve the following initial value problems.
   1. \( y' +(\cos(x))y=\cos(x) \), \( y(\pi)=0 \)
   2. \( (x^2+1)y'-2xy=x^2+1 \), \( y(1)=\pi \)
   
   
   
2. Convert the following systems of differential equations to an equivalent first order system (using \(x_1,x_2,\dots \) as dependent variable names). State whether the system is linear or not. If the system is linear, write the equivalent first order system in vector-matrix form: \({\bf x}'(t) = A(t){\bf x}+{\bf g}(t). \)
   1. \( (y'')^3+\sin(y'z')=e^{y+z''} \) and \( z'''+e^{2t}y'+\sin(t)z'=5 \)
   2. \( y'''+t^5y''+e^{-t^2}y'+\sin(t)y=t^3-7 \)
   
   
   
3. Find the general solution of the following homogeneous linear differential equations:
   1. \( y^{(5)}-4y'''=0 \)
   2. \( y^{(6)}+6y^{(5)}+16y^{(4)}+32y'''+48y''+32y'=0 \)
   3. \( y^{(4)}+4y'''+24y''+40y'+100y=0 \)
   4. \( x^3y'''+x^2y''-6xy'+6y=0 \)
   5. \( x^4y^{(4)}+10x^3y'''+32x^2y''+54xy'+36y=0 \)
   6. \( x^3y'''+8x^2y''+23xy'+13y=0 \)
   
   
   
4. (Rabenstein 5.5 #29 b,d,f page 207 in 2^nd edition and page 236 in 3^rd edition) Find a linear homogeneous differential equation with real constant coefficients, whose order is as low as possible, that has the given function as a solution.
   2. \( x-e^{3x} \)
   4. \( e^x\sin(2x) \)
   6. \( \cos(2x)+3e^{-x} \)
   Find a linear homogeneous differential equation of Cauchy-Euler type (with real coefficients), whose order is as low as possible, that has the given function as a solution.
   1. \( x^{-3}\ln(x)+x^5 \)
   2. \( x^{-1}\cos(3\ln(x)) \)
   
   
   
5. "Fun" with differential operators. For convenience, let \( D=\frac{d}{dx} \). Let \( a,b,c \) be constants.
   1. The differential operators \( x^2D \) and \( Dx^2 \) are not equal. Find an (operator) equation relating them.
   2. The equation \( (x^2D-a)[y]=0 \) is the same as \( x^2y'-ay=0 \). Find the general solution of this equation. Separation of variables should work.
   3. Find the second order linear equation equivalent to \( (x^2D-a)(x^2D-b)[y]=0 \). Find the general solution of such an equation. Consider 3 cases: (1) Distinct real roots, (2) A repeated real root, and (3) A pair of complex conjugate roots.
   4. Find the third order linear equation equivalent to \( (x^2D-a)^3[y]=0 \). Also, find the general solution of this equation.