Question

In each of problems 9 through 20: (a) find the solution of the given initial value problem in explicit form.

Answer 1

`find the solution of the inital value problem.\\</p><p>consider the differential equation y'=(x(x^2+1))/(4y^3)\\</p><p>with initial condition y(0)=-(1)/(2)\\</p><p>y'=(x(x^2+1))/(4y^3)\\</p><p>(dy)/(dx)=(x(x^2+1))/(4y^3)\\</p><p>4y^3dy=x(x^2+1)dx\\</p><p>taken integral\\</p><p>4y^3dy=x(x^2+1)dx\\</p><p>\int 4y^3dy=\int x(x^2+1)dx\\</p><p>(4y^4)/(4)=(x^4)/(3)+(x^2)/(2)+c\\</p><p>put x=0,and y(0)=-(-1)/(√(2))\\</p><p>y^4=(x^4)/(3)+(x^2)/(2)+c\\`
`((-1)/(√(2)))^4=0+0+c\\</p><p>c=(1)/(4)\\</p><p>therefore, the solution is\\</p><p>y^4=(x^4)/(3)+(x^2)/(2)+(1)/(4)\\</p><p>y=((x^4)/(3)+(x^2)/(2)+(1)/(4))^(1)/(4)`