Solve the i.v.p DE

Question

Solve the i.v.p $y^{(4)}-y'''=0 , y(0)=0, y'(0)=0, y"(0)=0, y"'(0)=0$

Would I use the formula $a^{(1/n)}=R^{(1/n)}e^{(e^{i(alpha+2k(\pi))/n})}$

Answer 1

There is no calculation needed. The family of solutions consists of linear combinations of $4$ functions. The initial conditions make all the constants $0$, so the solution is the identically $0$ function.

Another way of thinking about it is that the solution of the initial value problem is unique. But identically $0$ works, so it is the only solution.

Answer 2

I don't think so! In fact you should set an axillary equation regarding to the OE as $$m^4-m^3=0$$ and so solve it: $$m=0,~~~\text{3 times}~~~m=1$$ and then write the proper solution as $$y=C_1+C_2x+C_3x^2+C_4\text{e}^x$$ Now put the initial conditions. I don't think you get another one but the Andrea's one.