Chapter 2: Problem 1

Find all the solutions of the first-order differential equations. When an initial condition is given, find the particular solution satisfying that condition. a. \(\frac{d y}{d x}=\frac{e^{x}}{2 y}\). b. \(\frac{d y}{d t}=y^{2}\left(1+t^{2}\right), y(0)=1\). c. \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{x}\) d. \(x y^{\prime}=y(1-2 y), \quad y(1)=2\) e. \(y^{\prime}-(\sin x) y=\sin x\) f. \(x y^{\prime}-2 y=x^{2}, y(1)=1\) g. \(\frac{d s}{d t}+2 s=s t^{2}, \quad s(0)=1\) h. \(x^{\prime}-2 x=t e^{2 t}\) i. \(\frac{d y}{d x}+y=\sin x, y(0)=0\). j. \(\frac{d y}{d x}-\frac{3}{x} y=x^{3}, y(1)=4\)

Short Answer

Expert verified

The general solutions and particular solutions (when an initial condition is given) of the given first order differential equations have been found. Each equation involved a different type of first order differential equation, so a variety of methods were used, including separation of variables, integration, homogeneous solutions, and variation of parameters or undetermined coefficients for non-homogeneous equations.

Step by step solution

Exercise a

First of all, rearrange the equation into separable form, \(2y \, dy = e^{x} \, dx\). Now you need to integrate both sides. The integral form would be \(y^{2} = e^{x} + C\). Here \(C\) is the constant of integration.

Exercise b

First, we separate the variables, giving us \(\frac{1}{y^{2}} \, dy = (1 + t^{2}) \, dt\). We then integrate to get \(-\frac{1}{y} = t + \frac{t^{3}}{3}+ C\). With the initial condition \(y(0) = 1\), we solve for the constant to obtain the particular solution.

Exercise c

The given differential equation is in separated form. Its integral will be \(arcsin(y) = ln|x|+C\). \(C\) is the constant of integration.

Exercise d

The differential equation is separable, and the integrated form will be \(ln|y| - y^{2}= ln|x| + C\). We then use the initial condition \(y(1) = 2\) to find the particular solution.

Exercise e

The given differential equation is a first order linear homogeneous one. The solution will be \(y = e^{\int sin(x) \, dx}\). We then integrate inside the exponent.

Exercise f

This is a first order linear non-homogeneous differential equation. You can first solve the homogeneous differential equation and then use variation of parameters or the method of undetermined coefficients to solve for the particular solution. After that, you use the initial condition \(y(1) = 1\) to solve for the constant of integration.

Exercise g

This is a first order linear non-homogeneous differential equation similar to exercise f. We will use the same method to solve this equation.

Exercise h

This is a first order linear non-homogeneous differential equation. We can use similar methods to solve this as well.

Exercise i

This is a first order linear non-homogeneous differential equation. We can proceed the same way as in previous problems. Using the initial condition \(y(0) = 0\), we can find the particular solution.

Exercise j

This is also a first order linear non-homogeneous differential equation. Similar methods are used to solve this differential equation. The initial condition \(y(1) = 4\) is used to find the constant of integration in the particular solution.

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