Experiment 22.1
In each of the following, you are given a differential equation and a function y=f(x). Following the procedure of Section 22.1, check in each case whether the function is a solution to the differential equation.
1. x y'=3 y +x^4 cos x, y=x^3 sin x.
2. x y'=3 y +x^4 cos x, y=x^2 sin x.
3. x^2 y''-x y'+y=ln x, y=x ln x+ln x +2.
4. x^2 y''+x y'+y=0, y= cos(ln x) + sin(ln x).
5. x^3(y')^2+x^2 y y'+4=0, y=-(x+4)/x.
6. (y')^2+x y=e^x, y=e^x-e^-x.
1.
Clear[f, y, x, left, right]
f[x_] := x^3*Sin[x]
left[y_] := x*y'[x]
right[y_] := 3*y[x]
In[19]:= Simplify[left[f]]
Out[19]= left[f]
2.
Clear[f, y, x, left, right]
f[x_] := x^2*Sin[x]
left[y_] := x*y'[x]
right[y_] := 3*y + x^4*Cos[x]
In[13]:= Simplify[left[f]]
Out[13]= left[f]
3.
Clear[f, y, x, left, right]
f[x_] := x*Log[x] + Log[x] + 2
left[y_] := x^2*y''[x] - x*y'[x] + y[x]
right[y_] := Log[x]
[left[f]]
Log[x]
-
Clear[f, x, y, left, right]
f[x_] := Cos[Log[x]] + Sin[Log[x]]
left[y_] := x^2*y''[x] + x*y'[x] + y[x]
right[y_] := 0
In[11]:= [left[f]]
During evaluation of In[11]:= Syntax::tsntxi: "[left[f]]" is incomplete; more input is needed.
5.
Clear[f, x, y, left, right]
f[x_] := -[x + 4]/x
left[y_] := x^3*[y'[x]]^2 + x^2*y[x]*y'[x] + 4
right[y_] := 0
Simplify[left[f]]
left[f]
Clear[f, x, y, left, right]
f[x_] := E^[x] - E^[-x]
left[y_] := [y'[x]]^2 + x*y'[x]
right[y_] := E^[x]
Simplify[left[f]]
left[f]
