Graficas y ecuaciones- Manipulate

Manipulate[Plot[{Cos[x]+Cos[w x], Cos[w x]},{x,0,T}],{T,6,100},{w,0,2}]

DSolve

 DSolve[y'[x]+y[x]==a Sin[x],y[x],x]

Con condicion inicial

DSolve[{y'[x]==Cos[x] Sqrt[Abs[y[x]]],y[0]==0},y[x],x]

NDSolve[{y'[x]==Cos[x]*Sqrt[Abs[y[x]]],y[0]==0.1},y,{x,-2,2}]

Plot[Evaluate[y[x]/.%],{x,-2,2}]

Plano fase de Sistema no lineal 2D

Manipulate[Show[ParametricPlot[Evaluate[First[{x[t],y[t]}/.NDSolve[{x'[t]==y[t],y'[t]==-x[t]+(1-x[t]*x[t])*y[t],Thread[{x[0],y[0]}==point]},{x,y},{t,0,T}]]],{t,0,T},PlotStyle->Red]],{w,.5,2},{{T,20},1,100},{{point,{3,0}},Locator},SaveDefinitions->True]

Oscilador armonico con forzamiento

Manipulate[Show[ParametricPlot[Evaluate[First[{x[t],y[t]}/.NDSolve[{x'[t]==(-1+a Cos[t]*Cos[t])*x[t]+(1-a*Cos[t]*Sin[t])*y[t],y'[t]==(-1-a*Cos[t]*Sin[t])*x[t]+(-1+a Sin[t]*Sin[t])*y[t],Thread[{x[0],y[0]}==point]},{x,y},{t,0,T}]]],{t,0,T},PlotStyle->Red]],{a,.5,2},{{T,20},1,200},{{point,{3,0}},Locator},SaveDefinitions->True]

Manipulate[Show[Plot[Evaluate[First[{x[t],y[t]}/.NDSolve[{x'[t]==y[t],y'[t]==-x[t]+0.5 Cos[w t],Thread[{x[0],y[0]}==point]},{t,x},{t,0,T}]]],{t,0,T},PlotStyle->Red]],{w,.5,2},{{T,20},1,200},{{point,{3,0}},Locator},SaveDefinitions->True]