# Linear equations questions and answers

Recent questions in Second order linear equations
Second order linear equations

### Solve each of the differential equations in Table 1. Include the characterestic polynomial and its roots with your answer. $$\displaystyle{y}{''}-{3}{y}'+{2}{y}={0}$$

Second order linear equations

### Find the solution of $$\displaystyle{f}{''}{\left({x}\right)}={8}{x}+ \sin{{x}}$$

Second order linear equations

### We give linear equations y=1.5x

Second order linear equations

### We give linear equations y=-3x

Second order linear equations

### The integrating factor method, which was an effective method for solving first-order differential equations, is not a viable approach for solving second-order equstions. To see what happens, even for the simplest equation, consider the differential equation $$\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}$$. Lagrange sought a function $$\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}$$ such that if one multiplied the left-hand side of $$\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}$$ bu $$\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}$$, one would get $$\displaystyle\mu{\left({t}\right)}{\left[{y}{''}+{y}'+{y}\right]}={d}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}+{g{{\left({t}\right)}}}{y}\right]}$$ where g(t)g(t) is to be determined. In this way, the given differential equation would be converted to $$\displaystyle{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}'+{g{{\left({t}\right)}}}{y}\right]}=\mu{\left({t}\right)}{f{{\left({t}\right)}}}$$, which could be integrated, giving the first-order equation $$\displaystyle\mu{\left({t}\right)}{y}'+{g{{\left({t}\right)}}}{y}=\int\mu{\left({t}\right)}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}+{c}$$ which could be solved by first-order methods. (a) Differentate the right-hand side of $$\displaystyle\mu{\left({t}\right)}{\left[{y}{''}+{y}'+{y}\right]}={d}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}+{g{{\left({t}\right)}}}{y}\right]}$$ and set the coefficients of y,y' and y'' equal to each other to find g(t). (b) Show that the integrating factor $$\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}$$ satisfies the second-order homogeneous equation $$\displaystyle\mu{''}-\mu'+\mu={0}$$ called the adjoint equation of $$\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}$$. In other words, althought it is possible to find an "integrating factor" for second-order differential equations, to find it one must solve a new second-order equation for the integrating factor μ, which might be every bit as hard as the original equation. (c) Show that the adjoint equation of the general second-order linear equation $$\displaystyle{y}{''}+{p}{\left({t}\right)}{y}'+{q}{\left({t}\right)}{y}={f{{\left({t}\right)}}}$$ is the homogeneous equation $$\displaystyle\mu{''}-{p}{\left({t}\right)}\mu'+{\left[{q}{\left({t}\right)}-{p}'{\left({t}\right)}\right]}\mu={0}$$.

Second order linear equations

### Use the family in Problem 1 to ﬁnd a solution of $$y+y''=0$$ that satisﬁes the boundary conditions $$y(0)=0,y(1)=1.$$

Second order linear equations

### Solve the equation: $$\displaystyle{\left({a}-{x}\right)}{\left.{d}{y}\right.}+{\left({a}+{y}\right)}{\left.{d}{x}\right.}={0}$$

Second order linear equations

### Solve the equation: $$\displaystyle{\left({x}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}{\left({y}^{{2}}+{1}\right)}$$

Second order linear equations

### How to integrate $$\displaystyle{{\cos}^{{2}}{\left({2}{x}\right)}}$$?

Second order linear equations

### $$\displaystyle{\left({b}\right)}{y}{''}+{y}′+{3}⋅{a}{5}{y}={0}$$

Second order linear equations

### Solve the linear equations by considering y as a function of x, that is, $$y = y(x)$$. $$y'+2y=4$$

Second order linear equations

### Solve the correct answer linear equations by considering y as a function of x, that is, $$\displaystyle{y}={y}{\left({x}\right)}.\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+{y}= \cos{{x}}$$

Second order linear equations

### Solve differential equation: $$\displaystyle{y}'+{y}^{{2}}{\sin{{x}}}={0}$$

Second order linear equations

### Make and solve the given equation $$x\ dx\ +\ y\ dy=a^{2}\frac{x\ dy\ -\ y\ dx}{x^{2}\ +\ y^{2}}$$

Second order linear equations

### Solve the linear equations by considering y as a function of x, that is, y = y(x). $$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}-{y}={4}{e}^{x},{y}{\left({0}\right)}={4}$$

Second order linear equations

### Give the correct answer and solve the given equation: $$\displaystyle{y}\ \text{ - 4y}+{3}{y}={x},{y}_{{1}}={e}^{x}$$

Second order linear equations

### Solve the linear equations by considering y as a function of x, that is, $$y = y(x).$$ $$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}+{\left(\frac{1}{{x}}\right)}{y}={x}$$

Second order linear equations

### The coefficient matrix for a system of linear differential equations of the form $$\displaystyle{y}^{{{1}}}={A}_{{{y}}}$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system. $$\left[\lambda_{1}=-1\Rightarrow\left\{\begin{bmatrix}1 0 3 \end{bmatrix}\right\},\lambda_{2}=3i\Rightarrow\left\{\begin{bmatrix}2-i 1+i 7i \end{bmatrix}\right\},\lambda_3=-3i\Rightarrow\left\{\begin{bmatrix}2+i 1-i -7i \end{bmatrix}\right\}\right]$$

Second order linear equations