By Henri Cartan

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Extra resources for Differential Forms

Example text

We can solve all those first-order differential equations in which time does not appear explicitly. Now, suppose we have a differential equation of the form ely/dt= f(y / I), such as, for example, the equation ely / dt = sin(y / I). Differential equations of this form are called homogeneous equations. Since the right-hand side only depends on the single variable y / t. it suggests itself to make the substitution y /1 = V or y = tv. (a) Show that this substitution replaces the equation dy / dt = f(y / t) by the equivalent equation tdv/dt+v=f(v), which is separable.

The general solution of the differential equation must be left in the form e l + 3ry + siny = c since we cannot find y explicitly as a function of t from this equation. Second Method: From (ii), CP(t,y) = 3ty + siny + k(t). Differentiating this expression with respect to t, and using (i) we obtain that 3y+k'(t)=3y+e'. Thus, k(t)=e ' and CP(t,y)=3ty+siny+e '. Third Method: From (i) and (ii) cp(t,y) = e l + 3ty + h(y) and cp(t,y) =3ty +siny + k( t). Comparing these two expressions for the same function CP(t,y) it is obvious 51 1 First-order differential equations that h(y)=siny and k(t)=e l.

Differentiating this expression with respect to t and using (i) we obtain (t 4 + 4t 3 )e t +Y + k'( t) =4t 3e t +y + t 4e t +y + 2t. Thus, k(t)= t 2 and the general solution of the differential equation is (t,y)=t4e t +Y +y2+t 2=c. Setting t=O andy=l in this equation yields c = 1. Thus, the solution of our initial-value problem is defined implicitly by the equation t 4e l +Y + t 2+ y2 = 1. Suppose now that we are given a differential equation dy M(t,y)+ N(t,y) dt =0 (7) which is not exact. Can we make it exact?