First order differential equations (integrating factor)
5
The function \(y ( x )\) satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = x \ln x + \frac { y } { x }$$
and
$$y ( 1 ) = 1$$
Use the Euler formula
$$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\).
Use the formula
$$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with your answer to part (a)(i) to obtain an approximation to \(y ( 1.2 )\), giving your answer to three decimal places.
Show that \(\frac { 1 } { x }\) is an integrating factor for the first-order differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 1 } { x } y = x \ln x$$
Solve this differential equation, given that \(y = 1\) when \(x = 1\).
Calculate the value of \(y\) when \(x = 1.2\), giving your answer to three decimal places.