Find higher derivatives from equation

A question is this type if and only if it asks to find d³y/dx³ or d⁴y/dx⁴ by differentiating the given differential equation and expressing the result in terms of lower derivatives.

6 questions · Standard +0.3

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Edexcel F2 2021 January Q5

9 marks Challenging +1.2

5. Given that $$\left( 2 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 3 y$$

show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 1 } { \left( 2 - x ^ { 2 } \right) } \left( 2 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \left( 1 - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) - 5 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that $y = 3$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 }$ at $x = 0$
obtain a series solution for $y$ in ascending powers of $x$ with simplified coefficients, up to and including the term in $x ^ { 3 }$

Edexcel FP2 2011 June Q2

7 marks Standard +0.3

2. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { x } \left( 2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + 1 \right)$$

Show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \mathrm { e } ^ { x } \left[ 2 y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + k y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + 1 \right]$$ where $k$ is a constant to be found. Given that, at $x = 0 , y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2$,
find a series solution for $y$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.

OCR Further Additional Pure Specimen Q3

5 marks Standard +0.3

3 Given $z = x \sin y + y \cos x$, show that $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } + z = 0$.

Edexcel FP1 Specimen Q4

9 marks Challenging +1.2

4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$

Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a x \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$$ where $a$ and $b$ are integers to be found.
Hence find a series solution, in ascending powers of $x$, as far as the term in $x ^ { 5 }$, of the differential equation (I) where $y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$ at $x = 0$

Edexcel FP2 Q10

12 marks Standard +0.8

10. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } + y = 0$$

Find an expression for $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$. Given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$ at $x = 0$,
find the series solution for $y$, in ascending powers of $x$, up to an including the term in $x ^ { 3 }$.
Comment on whether it would be sensible to use your series solution to give estimates for $y$ at $x = 0.2$ and at $x = 50$.
[0pt] [P6 June 2002 Qn 4]

AQA Further Paper 2 2023 June Q1

1 marks Easy -1.8

1 Given that $y = \sin x + \sinh x$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y$ Circle your answer. $$\begin{array} { l l l l } 2 \sin x & - 2 \sin x & 2 \sinh x & - 2 \sinh x \end{array}$$