# Find the particular solution that satisfies the differential equation and the in

Find the particular solution that satisfies the differential equation and the initial condition. $$\displaystyle{f}”{\left({x}\right)}={\sin{{x}}},{f}’{\left({0}\right)}={1},{f{{\left({0}\right)}}}={6}$$

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escumantsu
Step 1
$$\displaystyle\text{We have: }\ {f}{''}{\left({x}\right)}={\sin{{x}}},{f}'{\left({0}\right)}={1},{f{{\left({0}\right)}}}={6}$$
$$\displaystyle\text{We integrate }\ {f}{''}{\left({x}\right)}={\sin{{x}}}\ \text{ (first) and }\ {f}'{\left({0}\right)}\ \text{ (second) to obtain a generalized solution:}$$
$$\displaystyle\int{f}{''}{\left({x}\right)}{\left.{d}{x}\right.}=\int{\sin{{x}}}{\left.{d}{x}\right.}=-{\cos{{x}}}+{C}_{{1}}$$
Step 2
Now, we are applying the initial condition $$\displaystyle{f}'{\left({0}\right)}={1}:$$
$$\displaystyle-{\cos{{0}}}+{C}_{{1}}={1}$$
$$\displaystyle-{1}+{C}_{{1}}={1}$$
$$\displaystyle{C}_{{1}}={1}+{1}$$
$$\displaystyle{C}_{{1}}={2}$$
$$\displaystyle\text{So: }\ {f}'{\left({x}\right)}=-{\cos{{x}}}+{2}$$
Now, we integrate f`(x):
$$\displaystyle\int{f}'{\left({x}\right)}{\left.{d}{x}\right.}=\int{\left(-{\cos{{x}}}+{2}\right)}{\left.{d}{x}\right.}$$
$$\displaystyle=\int-{\cos{{x}}}{\left.{d}{x}\right.}+\int{2}{\left.{d}{x}\right.}$$
$$\displaystyle=-{\sin{{x}}}+{2}{x}+{C}_{{2}}$$
Now, we are applying the initial condition f(0)=6:
$$\displaystyle{\sin{{0}}}+{2}\cdot{0}+{C}_{{2}}={6}$$
$$\displaystyle{0}+{C}_{{2}}={6}$$
$$\displaystyle{C}_{{2}}={6}$$
$$\displaystyle\text{So: }\ {f{{\left({x}\right)}}}=-{\sin{{x}}}+{2}{x}+{6}$$
Finaly, the particular solution is:
$$\displaystyle{f{{\left({x}\right)}}}=-{\sin{{x}}}+{2}{x}+{6}$$
Step 3
Explanetion:
$$\displaystyle{\left({1}\right)}:{f}{''}{\left({x}\right)}={\sin{{x}}}$$
$$\displaystyle{\left({2}\right)}:\int{\sin{{x}}}{\left.{d}{x}\right.}=-{\cos{{x}}}+{C}$$
$$\displaystyle{\left({3}\right)}:{f}'{\left({x}\right)}=-{\cos{{x}}}+{2}$$
$$\displaystyle{\left({4}\right)}:\int{\cos{{x}}}{\left.{d}{x}\right.}={\sin{{x}}}+{C}$$
Result
$$\displaystyle{f{{\left({x}\right)}}}=-{\sin{{x}}}+{2}{x}+{6}$$