Taylor series about x=1: differential equation with given conditions at x=1

Pre-U Pre-U 9795/1 2012 June Q8

11 marks Standard +0.8

8 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.

Determine $f ^ { \prime \prime } ( 1 )$.
Differentiate ( $*$ ) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.
Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.
Deduce, to 3 decimal places, an approximation for $\mathrm { f } ( 1.1 )$.

Pre-U Pre-U 9795/1 2016 Specimen Q8

10 marks Standard +0.8

8 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1 ,$$ and the conditions $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.

Determine $f ^ { \prime \prime } ( 1 )$.
Differentiate (*) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.
Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.
Deduce, to 3 decimal places, an approximation for $\mathrm { f } ( 1.1 )$.

Pre-U Pre-U 9795/1 2016 Specimen Q7

Standard +0.8

7 The function $f$ satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1 ,$$ and the conditions $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.

Determine $f ^ { \prime \prime } ( 1 )$.
Differentiate (*) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.
Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.
Deduce, to 3 decimal places, an approximation for $\mathrm { f } ( 1.1 )$.

Pre-U Pre-U 9795/1 2019 Specimen Q7

2 marks Challenging +1.2

7 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.

Determine $\mathrm { f } ^ { \prime \prime } ( 1 )$.
Differentiate (*) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.
Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.
Deduce, to 3 decimal places, an approximation for $\mathrm { f } ( 1.1 )$.

Pre-U Pre-U 9795/1 2020 Specimen Q7

2 marks Standard +0.8

7 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.

Determine $f ^ { \prime \prime } ( 1 )$.
Differentiate ( $*$ ) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.
Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.
Deduce, to 3 decimal places, an approximation for f(1.1).

Pre-U Pre-U 9795/1 2018 June Q7

6 marks Challenging +1.2

The function $y$ satisfies $\frac{d^2y}{dx^2} + x^2y = x$, and is such that $y = 1$ and $\frac{dy}{dx} = 1$ when $x = 1$.

Using the given differential equation
1. state the value of $\frac{d^2y}{dx^2}$ when $x = 1$, [1]
2. find, by differentiation, the value of $\frac{d^3y}{dx^3}$ when $x = 1$. [2]
Hence determine the Taylor series for $y$ about $x = 1$ up to and including the term in $(x-1)^3$ and deduce, correct to 4 decimal places, an approximation for $y$ when $x = 1.1$. [3]