7.1 Second-order linear equations  (Page 8/15)

$x^{3}y\text{″}+\left(x-1\right)y^{\prime }-8y=0$

$\left(1+y^2\right)y\text{″}+x{y}^{\prime }-3y=\text{cos}x$

$xy\text{″}+e^y{y}^{\prime }=x$

$y\text{″}+\frac{4}{x}{y}^{\prime }-8xy=5x^2+1$

$y\text{″}+\left(\text{sin}x\right)y^{\prime }-xy=4y$

$y\text{″}+\left(\frac{x+3}{y}\right)y^{\prime }=0$

[T] $y\text{″}+2y^{\prime }-3y=0;$ $y(x)=c_1{e}^x+c_2{e}^{\mathrm{-3}x}$

[T] $x^{2}y\text{″}-2y-3x^2+1=0;$ $y(x)=c_1{x}^2+c_2{x}^{\mathrm{-1}}+x^2\text{ln}(x)+\frac{1}{2}$

[T] $y\text{″}+14y^{\prime }+49y=0;$ $y(x)=c_1{e}^{\mathrm{-7}x}+c_{2}x{e}^{\mathrm{-7}x}$

[T] $6y\text{″}-49y^{\prime }+8y=0;$ $y(x)=c_1{e}^{x\text{/}6}+c_2{e}^{8x}$

Find the general solution to the linear differential equation.

$y\text{″}-3y^{\prime }-10y=0$

$y\text{″}-7y^{\prime }+12y=0$

$y\text{″}+4y^{\prime }+4y=0$

$4y\text{″}-12y^{\prime }+9y=0$

$2y\text{″}-3y^{\prime }-5y=0$

$3y\text{″}-14y^{\prime }+8y=0$

$y\text{″}+y^{\prime }+y=0$

$5y\text{″}+2y^{\prime }+4y=0$

$y\text{″}-121y=0$

$8y\text{″}+14y^{\prime }-15y=0$

$y\text{″}+81y=0$

$y\text{″}-y^{\prime }+11y=0$

$2y\text{″}=0$

$y\text{″}-6y^{\prime }+9y=0$

$3y\text{″}-2y^{\prime }-7y=0$

$4y\text{″}-10y^{\prime }=0$

$36\frac{d^{2}y}{d{x}^2}+12\frac{dy}{dx}+y=0$

$25\frac{d^{2}y}{d{x}^2}-80\frac{dy}{dx}+64y=0$

$\frac{d^{2}y}{d{x}^2}-9\frac{dy}{dx}=0$

$4\frac{d^{2}y}{d{x}^2}+8y=0$

$y\text{″}+5y^{\prime }+6y=0,y(0)=0,y^{\prime }(0)=\mathrm{-2}$

$y\text{″}+2y^{\prime }-8y=0,y(0)=5,y^{\prime }(0)=4$

$y\text{″}+4y=0,y(0)=3,y^{\prime }(0)=10$

$y\text{″}-18y^{\prime }+81y=0,y(0)=1,y^{\prime }(0)=5$

$y\text{″}-y^{\prime }-30y=0,y(0)=1,y^{\prime }(0)=\mathrm{-16}$

$4y\text{″}+4y^{\prime }-8y=0,y(0)=2,y^{\prime }(0)=1$

$25y\text{″}+10y^{\prime }+y=0,y(0)=2,y^{\prime }(0)=1$

$y\text{″}+y=0,y(\pi )=1,y^{\prime }(\pi )=\mathrm{-5}$

$y\text{″}+y^{\prime }-42y=0,y(0)=0,y(1)=2$

$9y\text{″}+y=0,y(\frac{3\pi }{2})=6,y(0)=\mathrm{-8}$

$y\text{″}+10y^{\prime }+34y=0,y(0)=6,y(\pi )=2$

$y\text{″}+7y^{\prime }-60y=0,y(0)=4,y(2)=0$

$y\text{″}-4y^{\prime }+4y=0,y(0)=2,y(1)=\mathrm{-1}$

$y\text{″}-5y^{\prime }=0,y(0)=3,y(\mathrm{-1})=2$

$y\text{″}+9y=0,y(0)=4,y\left(\frac{\pi }{3}\right)=\mathrm{-4}$

$4y\text{″}+25y=0,y(0)=2,y(2\pi )=\mathrm{-2}$

Find a differential equation with a general solution that is $y=c_1{e}^{x\text{/}5}+c_2{e}^{\mathrm{-4}x}.$

Find a differential equation with a general solution that is $y=c_1{e}^x+c_2{e}^{\mathrm{-4}x\text{/}3}.$

$y\text{″}+64y=0;y(0)=3,y^{\prime }(0)=16$

$y\text{″}-2y^{\prime }+10y=0y(0)=1,y^{\prime }(0)=13$

$y\text{″}+5y^{\prime }+15y=0y(0)=\mathrm{-2},y^{\prime }(0)=7$

(Principle of superposition) Prove that if $y_1(x)$ and $y_2(x)$ are solutions to a linear homogeneous differential equation, $y\text{″}+p(x)y^{\prime }+q(x)y=0,$ then the function $y(x)=c_1{y}_1(x)+c_2{y}_2(x),$ where $c_1$ and $c_2$ are constants, is also a solution.

Prove that if a, b, and c are positive constants, then all solutions to the second-order linear differential equation $ay\text{″}+b{y}^{\prime }+cy=0$ approach zero as $x\to \infty \text{.}$ ( Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)