$$[t{x}^{\prime }\left(t\right)+y\left(t\right)=0,t{y}^{\prime }\left(t\right)+x\left(t\right)=0]$$

$$[t{x}^{\prime }\left(t\right)+2x\left(t\right)=t,t{y}^{\prime }\left(t\right)-(2+t)x\left(t\right)-ty\left(t\right)=-t]$$

$$[t{x}^{\prime }\left(t\right)+2x\left(t\right)-2y\left(t\right)=t,t{y}^{\prime }\left(t\right)+x\left(t\right)+5y\left(t\right)=t^2]$$

$$[t^2(1-\sin \left(t\right))x^{\prime }\left(t\right)=t(1-2\sin \left(t\right))x\left(t\right)+t^{2}y\left(t\right),t^2(1-\sin \left(t\right))y^{\prime }\left(t\right)=(t\cos \left(t\right)-\sin \left(t\right))x\left(t\right)+t(1-t\cos \left(t\right))y\left(t\right)]$$

$$[x^{\prime }\left(t\right)+y^{\prime }\left(t\right)+y\left(t\right)=f\left(t\right),x^{\prime \prime }\left(t\right)+y^{\prime \prime }\left(t\right)+y^{\prime }\left(t\right)+x\left(t\right)+y\left(t\right)=g\left(t\right)]$$

$$[2x^{\prime }\left(t\right)+y^{\prime }\left(t\right)-3x\left(t\right)=0,x^{\prime \prime }\left(t\right)+y^{\prime }\left(t\right)-2y\left(t\right)={\mathrm{e}}^{2t}]$$

$$[x^{\prime }\left(t\right)-y^{\prime }\left(t\right)+x\left(t\right)=2t,x^{\prime \prime }\left(t\right)+y^{\prime }\left(t\right)-9x\left(t\right)+3y\left(t\right)=\sin \left(2t\right)]$$

$$[x^{\prime }\left(t\right)-x\left(t\right)+2y\left(t\right)=0,x^{\prime \prime }\left(t\right)-2y^{\prime }\left(t\right)=2t-\cos \left(2t\right)]$$

$$[t{x}^{\prime }\left(t\right)-t{y}^{\prime }\left(t\right)-2y\left(t\right)=0,t{x}^{\prime \prime }\left(t\right)+2x^{\prime }\left(t\right)+tx\left(t\right)=0]$$

$$[x^{\prime \prime }\left(t\right)+ay\left(t\right)=0,y^{\prime \prime }\left(t\right)-a^{2}y\left(t\right)=0]$$

$$[x^{\prime \prime }\left(t\right)=ax\left(t\right)+by\left(t\right),y^{\prime \prime }\left(t\right)=cx\left(t\right)+dy\left(t\right)]$$

$$[x^{\prime \prime }\left(t\right)=a_{1}x\left(t\right)+b_{1}y\left(t\right)+c_1,y^{\prime \prime }\left(t\right)=a_{2}x\left(t\right)+b_{2}y\left(t\right)+c_2]$$

$$[x^{\prime \prime }\left(t\right)+x\left(t\right)+y\left(t\right)=-5,y^{\prime \prime }\left(t\right)-4x\left(t\right)-3y\left(t\right)=-3]$$

$$[x^{\prime \prime }\left(t\right)=(3\cos {(at+b)}^2-1)c^{2}x\left(t\right)+\frac{3c^{2}y\left(t\right)\sin \left(2atb\right)}{2},y^{\prime \prime }\left(t\right)=(3\sin {(at+b)}^2-1)c^{2}y\left(t\right)+\frac{3c^{2}x\left(t\right)\sin \left(2atb\right)}{2}]$$

$$[x^{\prime \prime }\left(t\right)+6x\left(t\right)+7y\left(t\right)=0,y^{\prime \prime }\left(t\right)+3x\left(t\right)+2y\left(t\right)=2t]$$

$$[x^{\prime \prime }\left(t\right)-a{y}^{\prime }\left(t\right)+bx\left(t\right)=0,y^{\prime \prime }\left(t\right)+a{x}^{\prime }\left(t\right)+by\left(t\right)=0]$$

$$[a_1{x}^{\prime \prime }\left(t\right)+b_1{x}^{\prime }\left(t\right)+c_{1}x\left(t\right)-A{y}^{\prime }\left(t\right)=B{\mathrm{e}}^{i\omega t},a_2{y}^{\prime \prime }\left(t\right)+b_2{y}^{\prime }\left(t\right)+c_{2}y\left(t\right)+A{x}^{\prime }\left(t\right)=0]$$

$$[x^{\prime \prime }\left(t\right)+a(x^{\prime }\left(t\right)-y^{\prime }\left(t\right))+b_{1}x\left(t\right)=c_1{\mathrm{e}}^{i\omega t},y^{\prime \prime }\left(t\right)+a(y^{\prime }\left(t\right)-x^{\prime }\left(t\right))+b_{2}y\left(t\right)=c_2{\mathrm{e}}^{i\omega t}]$$

$$[\mathrm{a}11x^{\prime \prime }\left(t\right)+\mathrm{b}11x^{\prime }\left(t\right)+\mathrm{c}11x\left(t\right)+\mathrm{a}12y^{\prime \prime }\left(t\right)+\mathrm{b}12y^{\prime }\left(t\right)+\mathrm{c}12y\left(t\right)=0,\mathrm{a}21x^{\prime \prime }\left(t\right)+\mathrm{b}21x^{\prime }\left(t\right)+\mathrm{c}21x\left(t\right)+\mathrm{a}22y^{\prime \prime }\left(t\right)+\mathrm{b}22y^{\prime }\left(t\right)+\mathrm{c}22y\left(t\right)=0]$$

$$[x^{\prime \prime }\left(t\right)-2x^{\prime }\left(t\right)-y^{\prime }\left(t\right)+y\left(t\right)=0,y^{\prime \prime \prime }\left(t\right)-y^{\prime \prime }\left(t\right)+2x^{\prime }\left(t\right)-x\left(t\right)=t]$$

$$[x^{\prime \prime }\left(t\right)+y^{\prime \prime }\left(t\right)+y^{\prime }\left(t\right)=\sinh \left(2t\right),2x^{\prime \prime }\left(t\right)+y^{\prime \prime }\left(t\right)=2t]$$

$$[x^{\prime \prime }\left(t\right)-x^{\prime }\left(t\right)+y^{\prime }\left(t\right)=0,x^{\prime \prime }\left(t\right)+y^{\prime \prime }\left(t\right)-x\left(t\right)=0]$$

$$[x^{\prime }\left(t\right)=2x\left(t\right),y^{\prime }\left(t\right)=3x\left(t\right)-2y\left(t\right),z^{\prime }\left(t\right)=2y\left(t\right)+3z\left(t\right)]$$

$$[x^{\prime }\left(t\right)=4x\left(t\right),y^{\prime }\left(t\right)=x\left(t\right)-2y\left(t\right),z^{\prime }\left(t\right)=x\left(t\right)-4y\left(t\right)+z\left(t\right)]$$

$$[x^{\prime }\left(t\right)=y\left(t\right)-z\left(t\right),y^{\prime }\left(t\right)=x\left(t\right)+y\left(t\right),z^{\prime }\left(t\right)=x\left(t\right)+z\left(t\right)]$$

$$[x^{\prime }\left(t\right)-y\left(t\right)+z\left(t\right)=0,y^{\prime }\left(t\right)-x\left(t\right)-y\left(t\right)=t,z^{\prime }\left(t\right)-x\left(t\right)-z\left(t\right)=t]$$

$$[a{x}^{\prime }\left(t\right)=bc(y\left(t\right)-z\left(t\right)),b{y}^{\prime }\left(t\right)=ca(-x\left(t\right)+z\left(t\right)),c{z}^{\prime }\left(t\right)=ab(x\left(t\right)-y\left(t\right))]$$

$$[x^{\prime }\left(t\right)=cy\left(t\right)-bz\left(t\right),y^{\prime }\left(t\right)=az\left(t\right)-cx\left(t\right),z^{\prime }\left(t\right)=bx\left(t\right)-ay\left(t\right)]$$

$$[x^{\prime }\left(t\right)=h\left(t\right)y\left(t\right)-g\left(t\right)z\left(t\right),y^{\prime }\left(t\right)=f\left(t\right)z\left(t\right)-h\left(t\right)x\left(t\right),z^{\prime }\left(t\right)=x\left(t\right)g\left(t\right)-y\left(t\right)f\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)+y\left(t\right)-z\left(t\right),y^{\prime }\left(t\right)=y\left(t\right)+z\left(t\right)-x\left(t\right),z^{\prime }\left(t\right)=x\left(t\right)-y\left(t\right)+z\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-3x\left(t\right)+48y\left(t\right)-28z\left(t\right),y^{\prime }\left(t\right)=-4x\left(t\right)+40y\left(t\right)-22z\left(t\right),z^{\prime }\left(t\right)=-6x\left(t\right)+57y\left(t\right)-31z\left(t\right)]$$

$$[x^{\prime }\left(t\right)=6x\left(t\right)-72y\left(t\right)+44z\left(t\right),y^{\prime }\left(t\right)=4x\left(t\right)-4y\left(t\right)+26z\left(t\right),z^{\prime }\left(t\right)=6x\left(t\right)-63y\left(t\right)+38z\left(t\right)]$$

$$[x^{\prime }\left(t\right)=ax\left(t\right)+gy\left(t\right)+\beta z\left(t\right),y^{\prime }\left(t\right)=gx\left(t\right)+by\left(t\right)+\alpha z\left(t\right),z^{\prime }\left(t\right)=\beta x\left(t\right)+\alpha y\left(t\right)+cz\left(t\right)]$$

$$[t{x}^{\prime }\left(t\right)=2x\left(t\right)-t,t^3{y}^{\prime }\left(t\right)=-x\left(t\right)+t^{2}y\left(t\right)+t,t^4{z}^{\prime }\left(t\right)=-x\left(t\right)-t^{2}y\left(t\right)+t^{3}z\left(t\right)+t]$$

$$[at{x}^{\prime }\left(t\right)=bc(y\left(t\right)-z\left(t\right)),bt{y}^{\prime }\left(t\right)=ca(-x\left(t\right)+z\left(t\right)),ct{z}^{\prime }\left(t\right)=ab(x\left(t\right)-y\left(t\right))]$$

$$[x_1^{\prime }\left(t\right)=a{x}_2\left(t\right)+b{x}_3\left(t\right)\cos \left(ct\right)+b{x}_4\left(t\right)\sin \left(ct\right),x_2^{\prime }\left(t\right)=-a{x}_1\left(t\right)+b{x}_3\left(t\right)\sin \left(ct\right)-b{x}_4\left(t\right)\cos \left(ct\right),x_3^{\prime }\left(t\right)=-b{x}_1\left(t\right)\cos \left(ct\right)-b{x}_2\left(t\right)\sin \left(ct\right)+a{x}_4\left(t\right),x_4^{\prime }\left(t\right)=-b{x}_1\left(t\right)\sin \left(ct\right)+b{x}_2\left(t\right)\cos \left(ct\right)-a{x}_3\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-x\left(t\right)(x\left(t\right)+y\left(t\right)),y^{\prime }\left(t\right)=y\left(t\right)(x\left(t\right)+y\left(t\right))]$$

$$[x^{\prime }\left(t\right)=(ay\left(t\right)+b)x\left(t\right),y^{\prime }\left(t\right)=(cx\left(t\right)+d)y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)(a(px\left(t\right)+qy\left(t\right))+\alpha ),y^{\prime }\left(t\right)=y\left(t\right)(\beta +b(px\left(t\right)+qy\left(t\right)))]$$

$$[x^{\prime }\left(t\right)=h(a-x\left(t\right))(c-x\left(t\right)-y\left(t\right)),y^{\prime }\left(t\right)=k(b-y\left(t\right))(c-x\left(t\right)-y\left(t\right))]$$

$$[x^{\prime }\left(t\right)=y{\left(t\right)}^2-\cos \left(x\left(t\right)\right),y^{\prime }\left(t\right)=-y\left(t\right)\sin \left(x\left(t\right)\right)]$$

$$[x^{\prime }\left(t\right)=-x\left(t\right)y{\left(t\right)}^2+x\left(t\right)+y\left(t\right),y^{\prime }\left(t\right)=x{\left(t\right)}^{2}y\left(t\right)-x\left(t\right)-y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)+y\left(t\right)-x\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2),y^{\prime }\left(t\right)=-x\left(t\right)+y\left(t\right)-y\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2)]$$

$$[x^{\prime }\left(t\right)=-y\left(t\right)+x\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2-1),y^{\prime }\left(t\right)=x\left(t\right)+y\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2-1)]$$

$$[x^{\prime }\left(t\right)=-y\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2),y^{\prime }\left(t\right)=\{\begin{array}{cc}x{\left(t\right)}^2+y{\left(t\right)}^2& 2x\left(t\right)\le x{\left(t\right)}^2+y{\left(t\right)}^2\\ (\frac{x\left(t\right)}{2}-\frac{y{\left(t\right)}^2}{2x\left(t\right)})(x{\left(t\right)}^2+y{\left(t\right)}^2)& \mathrm{otherwise}\end{array}]$$

$$[x^{\prime }\left(t\right)=-y\left(t\right)+\left(\{\begin{array}{cc}x\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2-1)\sin \left(\frac{1}{x{\left(t\right)}^2+y{\left(t\right)}^2}\right)& x{\left(t\right)}^2+y{\left(t\right)}^2\ne 1\\ 0& \mathrm{otherwise}\end{array}\right),y^{\prime }\left(t\right)=x\left(t\right)+\left(\{\begin{array}{cc}y\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2-1)\sin \left(\frac{1}{x{\left(t\right)}^2+y{\left(t\right)}^2}\right)& x{\left(t\right)}^2+y{\left(t\right)}^2\ne 1\\ 0& \mathrm{otherwise}\end{array}\right)]$$

$$[(t^2+1)x^{\prime }\left(t\right)=-tx\left(t\right)+y\left(t\right),(t^2+1)y^{\prime }\left(t\right)=-x\left(t\right)-ty\left(t\right)]$$

$$[(x{\left(t\right)}^2+y{\left(t\right)}^2-t^2)x^{\prime }\left(t\right)=-2tx\left(t\right),(x{\left(t\right)}^2+y{\left(t\right)}^2-t^2)y^{\prime }\left(t\right)=-2ty\left(t\right)]$$

$$[{x^{\prime }\left(t\right)}^2+t{x}^{\prime }\left(t\right)+a{y}^{\prime }\left(t\right)-x\left(t\right)=0,x^{\prime }\left(t\right)y^{\prime }\left(t\right)+t{y}^{\prime }\left(t\right)-y\left(t\right)=0]$$

$$[x\left(t\right)=t{x}^{\prime }\left(t\right)+f(x^{\prime }\left(t\right),y^{\prime }\left(t\right)),y\left(t\right)=t{y}^{\prime }\left(t\right)+g(x^{\prime }\left(t\right),y^{\prime }\left(t\right))]$$

$$[x^{\prime \prime }\left(t\right)=a{\mathrm{e}}^{2x\left(t\right)}-{\mathrm{e}}^{-x\left(t\right)}+{\mathrm{e}}^{-2x\left(t\right)}\cos {\left(y\left(t\right)\right)}^2,y^{\prime \prime }\left(t\right)={\mathrm{e}}^{-2x\left(t\right)}\sin \left(y\left(t\right)\right)\cos \left(y\left(t\right)\right)-\frac{\sin \left(y\left(t\right)\right)}{\cos {\left(y\left(t\right)\right)}^3}]$$

$$[x^{\prime \prime }\left(t\right)=\frac{kx\left(t\right)}{{(x{\left(t\right)}^2+y{\left(t\right)}^2)}^{\frac{3}{2}}},y^{\prime \prime }\left(t\right)=\frac{ky\left(t\right)}{{(x{\left(t\right)}^2+y{\left(t\right)}^2)}^{\frac{3}{2}}}]$$

$$[x^{\prime }\left(t\right)=y\left(t\right)-z\left(t\right),y^{\prime }\left(t\right)=x{\left(t\right)}^2+y\left(t\right),z^{\prime }\left(t\right)=x{\left(t\right)}^2+z\left(t\right)]$$

$$[a{x}^{\prime }\left(t\right)=(b-c)y\left(t\right)z\left(t\right),b{y}^{\prime }\left(t\right)=(c-a)z\left(t\right)x\left(t\right),c{z}^{\prime }\left(t\right)=(a-b)x\left(t\right)y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)(y\left(t\right)-z\left(t\right)),y^{\prime }\left(t\right)=y\left(t\right)(-x\left(t\right)+z\left(t\right)),z^{\prime }\left(t\right)=z\left(t\right)(x\left(t\right)-y\left(t\right))]$$

$$[x^{\prime }\left(t\right)+y^{\prime }\left(t\right)=x\left(t\right)y\left(t\right),y^{\prime }\left(t\right)+z^{\prime }\left(t\right)=y\left(t\right)z\left(t\right),x^{\prime }\left(t\right)+z^{\prime }\left(t\right)=x\left(t\right)z\left(t\right)]$$

$$[x^{\prime }\left(t\right)=\frac{x{\left(t\right)}^2}{2}-\frac{y\left(t\right)}{24},y^{\prime }\left(t\right)=2x\left(t\right)y\left(t\right)-3z\left(t\right),z^{\prime }\left(t\right)=3x\left(t\right)z\left(t\right)-\frac{y{\left(t\right)}^2}{6}]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)(y{\left(t\right)}^2-z{\left(t\right)}^2),y^{\prime }\left(t\right)=y\left(t\right)(z{\left(t\right)}^2-x{\left(t\right)}^2),z^{\prime }\left(t\right)=z\left(t\right)(x{\left(t\right)}^2-y{\left(t\right)}^2)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)(y{\left(t\right)}^2-z{\left(t\right)}^2),y^{\prime }\left(t\right)=-y\left(t\right)(z{\left(t\right)}^2+x{\left(t\right)}^2),z^{\prime }\left(t\right)=z\left(t\right)(x{\left(t\right)}^2+y{\left(t\right)}^2)]$$

$$[x^{\prime }\left(t\right)=-x\left(t\right)y{\left(t\right)}^2+x\left(t\right)+y\left(t\right),y^{\prime }\left(t\right)=x{\left(t\right)}^{2}y\left(t\right)-x\left(t\right)-y\left(t\right),z^{\prime }\left(t\right)=y{\left(t\right)}^2-x{\left(t\right)}^2]$$

$$[(x\left(t\right)-y\left(t\right))(x\left(t\right)-z\left(t\right))x^{\prime }\left(t\right)=f\left(t\right),(-x\left(t\right)+y\left(t\right))(y\left(t\right)-z\left(t\right))y^{\prime }\left(t\right)=f\left(t\right),(-x\left(t\right)+z\left(t\right))(-y\left(t\right)+z\left(t\right))z^{\prime }\left(t\right)=f\left(t\right)]$$

$$[x_1^{\prime }\left(t\right)\sin \left(x_2\left(t\right)\right)=x_4\left(t\right)\sin \left(x_3\left(t\right)\right)+x_5\left(t\right)\cos \left(x_3\left(t\right)\right),x_2^{\prime }\left(t\right)=x_4\left(t\right)\cos \left(x_3\left(t\right)\right)-x_5\left(t\right)\sin \left(x_3\left(t\right)\right),x_3^{\prime }\left(t\right)+x_1^{\prime }\left(t\right)\cos \left(x_2\left(t\right)\right)=a,x_4^{\prime }\left(t\right)-(1-\lambda )a{x}_5\left(t\right)=-m\sin \left(x_2\left(t\right)\right)\cos \left(x_3\left(t\right)\right),x_5^{\prime }\left(t\right)+(1-\lambda )a{x}_4\left(t\right)=m\sin \left(x_2\left(t\right)\right)\sin \left(x_3\left(t\right)\right)]$$

$$[3x^{\prime }\left(t\right)+3x\left(t\right)+2y\left(t\right)={\mathrm{e}}^t,4x\left(t\right)-3y^{\prime }\left(t\right)+3y\left(t\right)=3t]$$

$$[x^{\prime }\left(t\right)=-3y\left(t\right),y^{\prime }\left(t\right)=2x\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-2y\left(t\right),y^{\prime }\left(t\right)=-4x\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-3x\left(t\right),y^{\prime }\left(t\right)=2y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=4y\left(t\right),y^{\prime }\left(t\right)=2y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right),y^{\prime }\left(t\right)=x\left(t\right)+2y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)-y\left(t\right),y^{\prime }\left(t\right)=x\left(t\right)+y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)+2y\left(t\right),y^{\prime }\left(t\right)=x\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-x\left(t\right)-2y\left(t\right),y^{\prime }\left(t\right)=2x\left(t\right)-y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-2x\left(t\right)-3y\left(t\right),y^{\prime }\left(t\right)=-x\left(t\right)+4y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-3y\left(t\right),y^{\prime }\left(t\right)=-2x\left(t\right)+y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-2x\left(t\right),y^{\prime }\left(t\right)=x\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-2x\left(t\right)-y\left(t\right),y^{\prime }\left(t\right)=-4y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=x\left(t\right)-2y\left(t\right),y^{\prime }\left(t\right)=-2x\left(t\right)+4y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=-6y\left(t\right),y^{\prime }\left(t\right)=6y\left(t\right)]$$

$$[x^{\prime }\left(t\right)=2x\left(t\right)+3y\left(t\right),y^{\prime }\left(t\right)=-x\left(t\right)-14]$$