Chapter 8: Problem 1
What is the order of \(y^{\prime \prime}(t)+9 y(t)=10 ?\)
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Determine whether the following equations are separable. If so, solve the initial value problem. $$\frac{d y}{d x}=e^{x-y}, y(0)=\ln 3$$
The equation \(y^{\prime}(t)+a y=b y^{p},\) where \(a, b,\) and \(p\) are real numbers, is called a Bernoulli equation. Unless \(p=1,\) the equation is nonlinear and would appear to be difficult to solve-except for a small miracle. By making the change of variables \(v(t)=(y(t))^{1-p},\) the equation can be made linear. Carry out the following steps. a. Letting \(v=y^{1-p},\) show that \(y^{\prime}(t)=\frac{y(t)^{p}}{1-p} v^{\prime}(t)\). b. Substitute this expression for \(y^{\prime}(t)\) into the differential equation and simplify to obtain the new (linear) equation \(v^{\prime}(t)+a(1-p) v=b(1-p),\) which can be solved using the methods of this section. The solution \(y\) of the original equation can then be found from \(v\).
Analysis of a separable equation Consider the differential equation \(y y^{\prime}(t)=\frac{1}{2} e^{t}+t\) and carry out the following analysis. a. Find the general solution of the equation and express it explicitly as a function of \(t\) in two cases: \(y>0\) and \(y < 0\) b. Find the solutions that satisfy the initial conditions \(y(-1)=1\) and \(y(-1)=2\) c. Graph the solutions in part (b) and describe their behavior as \(t\) increases. d. Find the solutions that satisfy the initial conditions \(y(-1)=-1\) and \(y(-1)=-2\) e. Graph the solutions in part (d) and describe their behavior as \(t\) increases.
A differential equation of the form \(y^{\prime}(t)=f(y)\) is said to be autonomous (the function \(f\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(f\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(y-3)(y+2)$$
Solve the following initial value problems. When possible, give the solution as an explicit function of \(t\) $$y^{\prime}(t)=\frac{\cos ^{2} t}{2 y}, y(0)=-2$$
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