Week 2

In [1]:

DSolve[t*x'[t]+x[t]==t*x[t]^2*Log[t],x[t],t]

DSolve[t*x'[t]+x[t]==t*x[t]^2*Log[t],x[t],t]

Out[1]:

In [2]:

Integrate[Log[t]/t,t]

Integrate[Log[t]/t,t]

Out[2]:

In [3]:

DSolve[x'[t] == (t*x[t]^2 + Sin[t])/(2*x[t]),x[t],t]

DSolve[x'[t] == (t*x[t]^2 + Sin[t])/(2*x[t]),x[t],t]

Out[3]:

In [4]:

Integrate[t*Sin[t],t]

Integrate[t*Sin[t],t]

Out[4]:

In [5]:

DSolve[y'[t] == t*y[t] - Sin[t], y[t],t]

DSolve[y'[t] == t*y[t] - Sin[t], y[t],t]

Out[5]:

In [6]:

DSolve[y'[t] == t*y[t], y[t],t]

DSolve[y'[t] == t*y[t], y[t],t]

Out[6]:

In [7]:

DSolve[Exp[t]+x[t]-1-(Cos[x[t]]-t)x'[t]==0,x[t],t]

DSolve[Exp[t]+x[t]-1-(Cos[x[t]]-t)x'[t]==0,x[t],t]

Out[7]:

In [8]:

$Assumptions = {t > 0};
FullSimplify[DSolve[x[t]^3+2*t*(t-x[t]^2)*x'[t]==0,x[t],t], Assumptions -> x \[Element] Reals]

$Assumptions = {t > 0}; FullSimplify[DSolve[x[t]^3+2*t*(t-x[t]^2)*x'[t]==0,x[t],t], Assumptions -> x \[Element] Reals]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out[8]:

In [10]:

DSolve[t*x'[t]+x[t]^2-x[t]==9t^2,x[t],t]

DSolve[t*x'[t]+x[t]^2-x[t]==9t^2,x[t],t]

Out[10]:

In [11]:

FullSimplify[Integrate[Sin[t]*Exp[-1/2*t^2],t]]

FullSimplify[Integrate[Sin[t]*Exp[-1/2*t^2],t]]

Out[11]:

In [12]:

DSolve[(x[t]^2-x[t])+t*x'[t]==0,x[t],t]

DSolve[(x[t]^2-x[t])+t*x'[t]==0,x[t],t]

Out[12]:

In [13]:

DSolve[p'[t]+p[t]/t==1/(t^2),p[t],t]

DSolve[p'[t]+p[t]/t==1/(t^2),p[t],t]

Out[13]:

In [14]:

DSolve[x^3*t'[x]+2t[x](t[x]-x^2)==0,t[x],x]

DSolve[x^3*t'[x]+2t[x](t[x]-x^2)==0,t[x],x]

Out[14]:

In [15]:

Integrate[t*Sin[t],t]

Integrate[t*Sin[t],t]

Out[15]:

In [16]:

DSolve[(1+x*y[x])*y[x]+(1-x*y[x])*x*y'[x]==0,y[x],x]

DSolve[(1+x*y[x])*y[x]+(1-x*y[x])*x*y'[x]==0,y[x],x]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out[16]:

In [17]:

DSolve[(x^2*y[x]^3+y[x])+(x^3*y[x]^2-x)*y'[x]==0,y[x],x]

DSolve[(x^2*y[x]^3+y[x])+(x^3*y[x]^2-x)*y'[x]==0,y[x],x]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out[17]: