$$y^{\prime \prime }+y=3x$$

$$y^{\prime \prime }-4y=12$$

$$y^{\prime \prime }-2y^{\prime }-3y=6$$

$$y^{\prime \prime }-2y^{\prime }+2y=2x$$

$$y^{\prime \prime }+2y=4$$

$$y^{\prime \prime }+2y=6x$$

$$y^{\prime \prime }+2y=6x+4$$

$$y^{\prime \prime }+16y={\mathrm{e}}^{3x}$$

$$y^{\prime \prime }-y^{\prime }-2y=3x+4$$

$$y^{\prime \prime }-y^{\prime }-6y=2\sin \left(3x\right)$$

$$4y^{\prime \prime }+4y^{\prime }+y=3x{\mathrm{e}}^x$$

$$y^{\prime \prime }+y^{\prime }+y=\sin {\left(x\right)}^2$$

$$2y^{\prime \prime }+4y^{\prime }+7y=x^2$$

$$y^{\prime \prime }-4y=\sinh \left(x\right)$$

$$y^{\prime \prime }-4y=\cosh \left(2x\right)$$

$$y^{\prime \prime }+2y^{\prime }-3y=1+x{\mathrm{e}}^x$$

$$y^{\prime \prime }+9y=2\cos \left(3x\right)+3\sin \left(3x\right)$$

$$y^{\prime \prime }+9y=2x^2{\mathrm{e}}^{3x}+5$$

$$y^{\prime \prime }-2y^{\prime }+2y={\mathrm{e}}^x\sin \left(x\right)$$

$$y^{\prime \prime }+4y=3x\cos \left(2x\right)$$

$$y^{\prime \prime }+3y^{\prime }+2y=x({\mathrm{e}}^{-x}-{\mathrm{e}}^{-2x})$$

$$y^{\prime \prime }-6y^{\prime }+13y=x{\mathrm{e}}^{3x}\sin \left(2x\right)$$

$$y^{\prime \prime }+4y=2x$$

$$y^{\prime \prime }+3y^{\prime }+2y={\mathrm{e}}^x$$

$$y^{\prime \prime }+9y=\sin \left(2x\right)$$

$$y^{\prime \prime }+y=\cos \left(x\right)$$

$$y^{\prime \prime }-2y^{\prime }+2y=1+x$$

$$y^{\prime \prime }+y^{\prime }+y=\sin \left(3x\right)\sin \left(x\right)$$

$$y^{\prime \prime }+9y=\sin {\left(x\right)}^4$$

$$y^{\prime \prime }+y=x\cos {\left(x\right)}^3$$

$$y^{\prime \prime }+3y^{\prime }+2y=4{\mathrm{e}}^x$$

$$y^{\prime \prime }-2y^{\prime }-8y=3{\mathrm{e}}^{-2x}$$

$$y^{\prime \prime }-4y^{\prime }+4y=2{\mathrm{e}}^{2x}$$

$$y^{\prime \prime }-4y=\sinh \left(2x\right)$$

$$y^{\prime \prime }+4y=\cos \left(3x\right)$$

$$y^{\prime \prime }+9y=\sin \left(3x\right)$$

$$y^{\prime \prime }+9y=2\sec \left(3x\right)$$

$$y^{\prime \prime }+y=\csc {\left(x\right)}^2$$

$$y^{\prime \prime }+4y=\sin {\left(x\right)}^2$$

$$y^{\prime \prime }-4y=x{\mathrm{e}}^x$$

$$x^{\prime \prime }+9x=10\cos \left(2t\right)$$

$$x^{\prime \prime }+4x=5\sin \left(3t\right)$$

$$x^{\prime \prime }+100x=225\cos \left(5t\right)+300\sin \left(5t\right)$$

$$x^{\prime \prime }+25x=90\cos \left(4t\right)$$

$$m{x}^{\prime \prime }+kx=F_0\cos \left(\omega t\right)$$

$$x^{\prime \prime }+4x^{\prime }+4x=10\cos \left(3t\right)$$

$$x^{\prime \prime }+3x^{\prime }+5x=-4\cos \left(5t\right)$$

$$2x^{\prime \prime }+2x^{\prime }+x=3\sin \left(10t\right)$$

$$x^{\prime \prime }+3x^{\prime }+3x=8\cos \left(10t\right)+6\sin \left(10t\right)$$

$$x^{\prime \prime }+4x^{\prime }+5x=10\cos \left(3t\right)$$

$$x^{\prime \prime }+6x^{\prime }+13x=10\sin \left(5t\right)$$

$$x^{\prime \prime }+6x^{\prime }+13x=10\sin \left(5t\right)$$

$$x^{\prime \prime }+2x^{\prime }+26x=600\cos \left(10t\right)$$

$$x^{\prime \prime }+8x^{\prime }+25x=200\cos \left(t\right)+520\sin \left(t\right)$$

$$y^{\prime \prime }-5y^{\prime }+6y=2{\mathrm{e}}^t$$

$$y^{\prime \prime }-y^{\prime }-2y=2{\mathrm{e}}^{-t}$$

$$y^{\prime \prime }+2y^{\prime }+y=3{\mathrm{e}}^{-t}$$

$$4y^{\prime \prime }-4y^{\prime }+y=16{\mathrm{e}}^{\frac{t}{2}}$$

$$y^{\prime \prime }+y=\tan \left(t\right)$$

$$y^{\prime \prime }+9y=9\sec {\left(3t\right)}^2$$

$$y^{\prime \prime }+4y^{\prime }+4y=\frac{{\mathrm{e}}^{-2t}}{t^2}$$

$$y^{\prime \prime }+4y=3\csc \left(2t\right)$$

$$y^{\prime \prime }+y=2\sec \left(\frac{t}{2}\right)$$

$$y^{\prime \prime }-2y^{\prime }+y=\frac{{\mathrm{e}}^t}{t^2+1}$$

$$y^{\prime \prime }-5y^{\prime }+6y=g\left(t\right)$$

$$y^{\prime \prime }+4y=g\left(t\right)$$

$$u^{\prime \prime }+\frac{u^{\prime }}{8}+4u=3\cos \left(\frac{t}{4}\right)$$

$$u^{\prime \prime }+\frac{u^{\prime }}{8}+4u=3\cos \left(2t\right)$$

$$u^{\prime \prime }+\frac{u^{\prime }}{8}+4u=3\cos \left(6t\right)$$

$$y^{\prime \prime }+{\omega }^{2}y=\cos \left(2t\right)$$

$$y^{\prime \prime }-2y^{\prime }+2y={\mathrm{e}}^{-t}$$

$$y^{\prime \prime }+4y=\{\begin{array}{cc}1& 0\le t<\pi \\ 0& \pi \le t<\mathrm{\infty }\end{array}$$

$$y^{\prime \prime }+4y=\{\begin{array}{cc}1& 0\le t<1\\ 0& 1\le t<\mathrm{\infty }\end{array}$$

$$y^{\prime \prime }+y=\{\begin{array}{cc}t& 0\le t<1\\ 2-t& 1\le t<2\\ 0& 2\le t<\mathrm{\infty }\end{array}$$

$$y^{\prime \prime }+y=\{\begin{array}{cc}1& 0\le t<3\pi \\ 0& 3\pi \le t<\mathrm{\infty }\end{array}$$

$$y^{\prime \prime }+2y^{\prime }+2y=\{\begin{array}{cc}1& \pi \le t<2\pi \\ 0& \mathrm{otherwise}\end{array}$$

$$y^{\prime \prime }+4y=\sin \left(t\right)-\mathrm{Heaviside}(t-2\pi )\sin \left(t\right)$$

$$y^{\prime \prime }+3y^{\prime }+2y=\{\begin{array}{cc}1& 0\le t<10\\ 0& \mathrm{otherwise}\end{array}$$

$$y^{\prime \prime }+y^{\prime }+\frac{5y}{4}=t-\mathrm{Heaviside}(t-\frac{\pi }{2})(t-\frac{\pi }{2})$$

$$y^{\prime \prime }+y^{\prime }+\frac{5y}{4}=\{\begin{array}{cc}\sin \left(t\right)& 0\le t<\pi \\ 0& \mathrm{otherwise}\end{array}$$

$$y^{\prime \prime }+4y=\mathrm{Heaviside}(t-\pi )-\mathrm{Heaviside}(t-3\pi )$$

$$u^{\prime \prime }+\frac{u^{\prime }}{4}+u=k(\mathrm{Heaviside}(t-\frac{3}{2})-\mathrm{Heaviside}(t-\frac{5}{2}))$$

$$u^{\prime \prime }+\frac{u^{\prime }}{4}+u=\frac{\mathrm{Heaviside}(t-\frac{3}{2})}{2}-\frac{\mathrm{Heaviside}(t-\frac{5}{2})}{2}$$

$$u^{\prime \prime }+\frac{u^{\prime }}{4}+u=\frac{\mathrm{Heaviside}(t-5)(t-5)-\mathrm{Heaviside}(t-5-k)(t-5-k)}{k}$$

$$y^{\prime \prime }+2y^{\prime }+2y=\delta (t-\pi )$$

$$y^{\prime \prime }+4y=\delta (t-\pi )-\delta (t-2\pi )$$

$$y^{\prime \prime }+3y^{\prime }+2y=\delta (t-5)+\mathrm{Heaviside}(t-10)$$

$$y^{\prime \prime }+2y^{\prime }+3y=\sin \left(t\right)+\delta (t-3\pi )$$

$$y^{\prime \prime }+y=\delta (t-2\pi )\cos \left(t\right)$$

$$y^{\prime \prime }+4y=2\delta (t-\frac{\pi }{4})$$

$$y^{\prime \prime }+2y^{\prime }+2y=\cos \left(t\right)+\delta (t-\frac{\pi }{2})$$

$$y^{\prime \prime }+\frac{y^{\prime }}{2}+y=\delta (-1+t)$$

$$y^{\prime \prime }+\frac{y^{\prime }}{4}+y=\delta (-1+t)$$

$$y^{\prime \prime }+y=\frac{\mathrm{Heaviside}(t-4+k)-\mathrm{Heaviside}(t-4-k)}{2k}$$

$$y^{\prime \prime }+2y^{\prime }+2y=f\left(t\right)$$

$$y^{\prime \prime }+2y^{\prime }+2y=\delta (t-\pi )$$

$$y^{\prime \prime }-3y^{\prime }+2y=\frac{1}{1+{\mathrm{e}}^{-x}}$$

$$y^{\prime \prime }-2y^{\prime }+y=7x^{\frac{3}{2}}{\mathrm{e}}^x$$

$$y^{\prime \prime }-2y^{\prime }+2y={\mathrm{e}}^x\sec \left(x\right)$$

$$y^{\prime \prime }+9y=\tan \left(3x\right)$$