数値微分方程式の役割

今日コンピュータの歴史に関する記事を読みました。最初の電子計算機は微分方程式を解くために作成されました。　
What I am wondering is why numerical differential equation solving is necessary. Couldn't the equations be solved analytically and the problems be solved by substituting into these algebraic solutions. Are some differential equations used in science unsolvable analytically? It seems that the solutions are often expressed as integrals, so that using even the non-differential expressions involves integration. But wouldn't the same integrals be used repeatedly, making the process simply one of substituting numerical values for results of the algebraic integration? I do not see why integration need be done more than once and why the numerical method to solve a differential equation need be used at all.
One partial differential equation often cited is the heat equation, du/dt = 0.5 d^2u/dx^2. The solution to this is:
u(x,t)=INTEGRAL(0,1)(g(x-y,t)f(y)dy, where
g(x,t) = SIGMA(-infinity, infinity)e**(-2x**2n**2t)e**(2pi(i)nx))
A commonly cited application of numerical integrators is in the computation of trajectories.
Two examples of problems which seem to be unsolvable via analytic methods are the Lorenz attractor and Newton's equations of gravitation for three or more particles.
The Lorenz attractor equations are:
dx/dt = -10(x-y)
dy/dt = -xz+28x-y
dz/dt = xy-8z/3

In the realm of finance, two well-known differential equations are the Black-Scholes differential equation and the Hull-White differential equation.
The Black-Scholes differential equation has a closed-form solution.
for:
Cr = theta + rs(delta) + 0.5(sigma**2)S**2(gamma), the solution is
C = delta(S) - e**(-r(T-t))B
This is somewhat misleading, as delta itself is a partial derivative of C with respect to S.
The Hull-White interest rate model is:
dr(t) = (theta(t)-alpha(t)r(t))dt + sigma(t)dW(t)
An expression for r (not really closed form, as it is in terms of theta) is
r(t) = e**(-alpha*t)r(0)+ (theta/alpha)*(1-e**(-alpha*t)+ sigma*e**(-alpha*t)*Integral(0,t)e**(alpha*u)dW(u)