Obtain the general solution of the different equation
$$\frac{d^2y}{dx^2}-3\frac{dy}{dx}+2y=20\sin x.$$
Hence find the particular solution for which : $y=0$, $\frac{dy}{dx}=0$ when $x=0$.
$\endgroup$ 91 Answer
$\begingroup$To find the particular solution, you simply take your general solution and plug in the values that you are given for the particular solution.
Your general solution is
$$y=Ae^x+Be^{2x}+2\sin x+6\cos x.$$
You have given that the particular solution has the properties $y(0)=0$ and $\frac{dy}{dx}(0)=0$. The first condition means that when $x=0$, then $y=0$, so plug $x=0$ into your general solution and set it equal to $0$:
$$0 = A+B+6.$$
To use the second condition, you first need to differentiate your general solution:
$$\frac{dy}{dx}=Ae^x+2Be^{2x} + 2\cos(x) - 6\sin(x),$$
and then the second condition says that when $x=0$, then $\frac{dy}{dx}=0$, so you can plug in $x=0$, and set it equal to $0$:
$$0 = A+2B+2.$$
What you have now is two equations in the two unknowns $A$ and $B$, which you can solve. Take the resulting values for $A$ and $B$ and plug them back into your general solution: You now have the particular solution you were looking for.
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