Find the particular solution that satisfies the differential equation and the initial condition. f”(x)=sin⁡x,f′(0)=1,f(0)=6

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escumantsu · Accepted Answer

Step 1We have:  f″(x)=sin⁡x,f′(0)=1,f(0)=6We integrate  f″(x)=sin⁡x  (first) and  f′(0)  (second) to obtain a generalized solution:∫f″(x)dx=∫sin⁡xdx=−cos⁡x+C1Step 2Now, we are applying the initial condition f′(0)=1:−cos⁡0+C1=1−1+C1=1C1=1+1C1=2So:  f′(x)=−cos⁡x+2Now, we integrate f`(x):∫f′(x)dx=∫(−cos⁡x+2)dx=∫−cos⁡xdx+∫2dx=−sin⁡x+2x+C2Now, we are applying the initial condition f(0)=6:sin⁡0+2⋅0+C2=60+C2=6C2=6So:  f(x)=−sin⁡x+2x+6Finaly, the particular solution is:f(x)=−sin⁡x+2x+6Step 3Explanetion:(1):f″(x)=sin⁡x(2):∫sin⁡xdx=−cos⁡x+C(3):f′(x)=−cos⁡x+2(4):∫cos⁡xdx=sin⁡x+CResultf(x)=−sin⁡x+2x+6

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

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