Systems of Linear Equations - Learn the Basics

Find Laplace Transform of each following
(a) ${\displaystyle t^n}$ , (b) $cos\omega t$

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

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 ${\displaystyle y{}^{″}+3y^{\prime }+2y=f\left(t\right)}$. Lagrange sought a function ${\displaystyle \mu \left(t\right)\mu \left(t\right)}$ such that if one multiplied the left-hand side of ${\displaystyle y{}^{″}+3y^{\prime }+2y=f\left(t\right)}$ bu ${\displaystyle \mu \left(t\right)\mu \left(t\right)}$, one would get ${\displaystyle \mu \left(t\right)[y{}^{″}+y^{\prime }+y]=d\frac{d}{dt}[\mu \left(t\right)y+g\left(t\right)y]}$ where g(t)g(t) is to be determined. In this way, the given differential equation would be converted to ${\displaystyle \frac{d}{dt}[\mu \left(t\right)y^{\prime }+g\left(t\right)y]=\mu \left(t\right)f\left(t\right)}$, which could be integrated, giving the first-order equation ${\displaystyle \mu \left(t\right)y^{\prime }+g\left(t\right)y=\int \mu \left(t\right)f\left(t\right)dt+c}$ which could be solved by first-order methods. (a) Differentate the right-hand side of ${\displaystyle \mu \left(t\right)[y{}^{″}+y^{\prime }+y]=d\frac{d}{dt}[\mu \left(t\right)y+g\left(t\right)y]}$ and set the coefficients of y,y' and y'' equal to each other to find g(t). (b) Show that the integrating factor ${\displaystyle \mu \left(t\right)\mu \left(t\right)}$ satisfies the second-order homogeneous equation ${\displaystyle \mu {}^{″}-{\mu }^{\prime }+\mu =0}$ called the adjoint equation of ${\displaystyle y{}^{″}+3y^{\prime }+2y=f\left(t\right)}$. 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 ${\displaystyle y{}^{″}+p\left(t\right)y^{\prime }+q\left(t\right)y=f\left(t\right)}$ is the homogeneous equation ${\displaystyle \mu {}^{″}-p\left(t\right){\mu }^{\prime }+[q\left(t\right)-p^{\prime }\left(t\right)]\mu =0}$.

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

${\displaystyle \left(b\right)y+y\prime +3.25=0}$

Solve the linear equations by considering y as a function of x, that is, $y=y(x)$.
$y^{\prime }+2y=4$

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