Answer to Question #312268 in Differential Equations for Abhishek sahu

Question

Solve y’=y-x2

, y(0)=1, by Picord’s method upto the third approximation . Hence find

the value of y(0,1), y(0,2)

Answer 1

$y\left( t,0 \right) =1\\y\left( t,1 \right) =y\left( 0 \right) +\int_0^t{\left( y\left( s,0 \right) -s^2 \right) ds}=1-\frac{t^3}{3}\\y\left( t,2 \right) =y\left( 0 \right) +\int_0^t{\left( y\left( s,1 \right) -s^2 \right) ds}=1+t-\frac{t^4}{12}-\frac{t^3}{3}\\y\left( t,3 \right) =y\left( 0 \right) +\int_0^t{\left( y\left( s,2 \right) -s^2 \right) ds}=1+t+\frac{t^2}{2}-\frac{t^5}{60}-\frac{t^4}{12}\\y\left( 0,1 \right) =y\left( 0,2 \right) =1$