# allen_83

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1. ## solve a non homogeneous differential equation

alright, wanted to give "the tree"'s idea a try. got no clue. am stuck. this is a type : $x'' = f(t,x,x')$ with both x and t missing if I were to think of it as a non-linear DE. how ?
2. ## solve a non homogeneous differential equation

That's right. This ODE is non-linear. The problem is as it is. $x''(t) = e^{ x(t)} - 1 for x'(0)= 0 and x(0) = 1$ Attempts: $x''(t) + 1 = e^{x(t)}; \lambda^2+1 = 0 \Rightarrow \lambda_{1} = i, \lambda_{2} = -i ; x(t) = D_{1}cos(t) + D_{2}sin(t) \because x(0) = 1 , x'(0) = 0$ solvin for D1 and D2 : D1 = 1, D2 = 0 and pluging them back in : $x(t) = cos(t)$ $x_{nonhom}(t) = A\cdot e^x - \dfrac{x^2}{2}$ $x'_{nonhom}(t) = A\cdot e^x - x$ $x''_{nonhom}(t) = A\cdot e^x - 1$ $A\cdot e^x - 1 + 1 = e^x \Rightarrow A = 1$ $x_{nonhom}(t) = e^x - \dfrac{x^2}{2}$ $x = x_{nonhom}(t) + x_{hom}(t)$ $x = e^{x(t)} - \dfrac{x(t)^2}{2} + cos(t)$ correct me if I'm wrong.
3. ## solve a non homogeneous differential equation

$A\cdot e^x - \dfrac {1}{2} x^2$ (right?) however, considering the general form : $x'' + ax' + bx = g(y)$ there is no place for a constant c. So I wonder if I'm actually allowed to move -1 to the left hand side of the equation(?)
4. ## solve a non homogeneous differential equation

I need a hint on how to solve the following DE. x'' = ex -1. The general answer to the equation will be the sum of inhom. + hom. solutions of the equation. Any help is appreciated.
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