# Simplifying solution of a differential equation

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Aleem Andrew on 14 Jan 2021
Commented: Aleem Andrew on 14 Jan 2021
The most simplified version of ySol(t), the solution to the differential equation below, is 1.5*sin(2t+0.7297), but the output of the following code is in terms of exponential functions. Can someone explain how the output can be further simplified?
syms y(t) m k
Dy = diff(y,t); Dy2 = diff(y,t,2);
ode = m*Dy2 + k*y == 0;
cond = [y(0) == 1,Dy(0) == sqrt(5)];
ySol(t) = dsolve(ode,cond)
ySol(t) = simplify(ySol(t),'steps',500)
pretty(ySol(t))
Aleem Andrew on 14 Jan 2021
That is because there is an additional equation relating k and m, sqrt(k/m) = 2, that I tried to include in the dsolve command to solve the system but got an error message when trying to solve a system of equations, [ode sqrt(k/m) == 2]. Instead the ode = m*Dy2 + k*y == 0; line can be modified to ode = (k/4)*Dy2 + k*y == 0; to obtain the numeric solution.

Walter Roberson on 14 Jan 2021
m = rand(); k = rand();
syms y(t)
Dy = diff(y,t);
Dy2 = diff(y,t,2);
ode = m*Dy2 + k*y == 0;
cond = [y(0) == 1,Dy(0) == sqrt(5)];
ySol(t) = dsolve(ode,cond)
ySol(t) =
ySol(t) = simplify(ySol(t),'steps',500)
ySol(t) =
pretty(ySol(t))
/ sqrt(43198488722811199054095930230) t \ sqrt(8639697744562239810819186046) sin| ------------------------------------- | 5 / sqrt(43198488722811199054095930230) t \ \ 157178273090335 / cos| ------------------------------------- | + --------------------------------------------------------------------------------- \ 157178273090335 / 274837532398538
vpa(ySol(t), 5)
ans =
Aleem Andrew on 14 Jan 2021

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