Obtain the general solution using the method of Undetermined Coefficients. where $$ is the deflection at the cantilever tip: $=y(L),$ and it is an unknown.

$($a$)$ Show that this equation leads to an ODE of the type:

$y^{\text{'}\text{'}}+k^{2}y=k^2$

and find the value of $k.$

$($b$)$ Find the general solution of the ODE as listed in $($a$).$ Hints: take advantage of the

boundary conditions: $y(0)=0,y^{\text{'}}(0)=0.$

$($c$)$ From the solution obtained in $($b$),$ determine load $P($critical loads$),$ such that $y(L)=$

$(0).$

$($d$)$ Suppose $=1,$ plot the first three buckling mode shapes of the beam, i$.$e$.,$ plot $y(x)$

under the first three minimum values of $P_c$ obtained in $($c$).$

Figure $1$: Buckling of a cantilever beam.

Solve the initial value problems.

$y_1^{\text{'}}=y_2-5sint$

$y_2^{\text{'}}=-4y_1+17cost$

$y_1(0)=5,y_2(0)=2$

$y^{\text{'}\text{'}}+y^{\text{'}}=4x{e}^x+3sinx$

Obtain the general solution using the method of Undetermined Coefficients.

$y^{\text{'}\text{'}}+y=cosx$

Obtain the general solution using the method of Variation of Parameters.

$x^2{y}^{\text{'}\text{'}}-2x{y}^{\text{'}}+2y=x^{3}sinx$

Solve the Bernoulli equation, known as the logistic equation $($see class note $03).$

$y^{\text{'}}=Ay-B{y}^2$

Obtain the particular solution for a mass$-$spring system

$y^{\text{'}\text{'}}+2{}_0{y}^{\text{'}}+{}_0^{2}y=f(t)$

under different $f(t)$ as listed below:

$($a$)f(t)=F$

$($b$)f(t)=Ft$

$($c$)f(t)=$Fsin$(t)$

where $F$ and $$ are constants, $F>0,>0.$

Consider a cantilever beam of length $L$ as shown in Figure $1,$ made of a material with

Young's modulus $E$ and a uniform cross$-$section whose moment of inertia is I. The

beam is subjected to a compressive load $P.$ To seek conditions under which the beam

will buckle, i$.$e$.,$ the beam can be in equilibrium under the load $P$ in a configuration

involving deflections in the $y$ direction, according to the Euler$-$Bernoulli beam theory,

$EI{y}^{\text{'}\text{'}}(x)=P[-y(x)]$