Derivatives from a nonlinear differential identity

The function y=f(x)y=f(x) satisfies the differential equation 2ydydx=y2−12y\frac{dy}{dx}=y^2-1 and its graph passes the point (0,3)(0,3).

Show that when x=0x=0, d2ydx2=2027\frac{d^2y}{dx^2}=\frac{20}{27}. marks

Find the Maclaurin’s series of yy in ascending powers of xx up to and including the term in x3x^3. marks

[2008/RJC/Prelim /II/5(c)]

Total marks: 6

Solution

Second derivative

Implicit differentiation

2ydydx=y2−12y\dfrac{dy}{dx}=y^2-1 with y(0)=3y(0)=3. Show that d2ydx2=2027\dfrac{d^2y}{dx^2}=\dfrac{20}{27} when x=0x=0.

Hint

  1. Find y′(0)y'(0) from the equation.

  2. Differentiate the equation (product rule) and substitute.

Answer · Second derivative

y′′(0)=20/27y\prime\prime(0)=20/27.

Maclaurin series

Maclaurin's theorem

Find the Maclaurin series of yy up to and including x3x^3.

Explore this idea

Hint

  1. Differentiate yy′′+(y′)2=yy′yy''+(y')^2=yy' once more.

  2. Find y′′′(0)y'''(0).

  3. Apply Maclaurin's theorem.

Answer · Maclaurin series

3+4x/3+10x2/27+14x3/243+O(x4)3+4x/3+10x^2/27+14x^3/243+O(x^4).

Explore the idea

The solution and its series

Compare the function with a finite polynomial. Change the highest included power and test x, then use the error plot to see how the approximation changes at that x.

Highest included powerxⁿ, n = 3
Test value xx = 0.50
Reset the controls
Function
3.76693
P₃(x)
3.76646
Error
-0.00047

No convergence interval is specified for this plot. Compare the measured errors.

Teal marks the function, amber marks the polynomial, and the red segment joins their values at the selected x.

Does keeping more powers help at this x?

Highest included power n runs horizontally; signed error Pₙ(x) − f(x) runs vertically. Zero means agreement at this x. A missing power leaves the polynomial unchanged; individual steps need not improve the error.

  • Function
  • Series truncated at the chosen power
Key idea

The coefficients come from the DE without solving it; y2=1+8exy^2=1+8e^x confirms them.

Ideas to try
Raise the degree: the polynomial hugs the solution more closely near 0.