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solniwko [45]
3 years ago
5

Find the general solution of dz/dt=5ze^(sint)cost+2e^(sint)cost

Mathematics
1 answer:
gizmo_the_mogwai [7]3 years ago
4 0

Answer:

The general solution of given differential equation is \frac{1}{5}ln|5z+2|=e^{\sin t}+C.

Step-by-step explanation:

The given differential equation is

\frac{dz}{dt}=5ze^{\sin t}\cos t+2e^{\sin t}\cos t

Taking out common factors.

\frac{dz}{dt}=(5z+2)e^{\sin t}\cos t

Using variable separable method, we get

\frac{dz}{5z+2}=e^{\sin t}\cos t dt

Integrate both sides.

\int\frac{dz}{5z+2}=\int e^{\sin t}\cos t dt

I_1=I_2         .... (1)

First solve LHS,

I_1=\int\frac{dz}{5z+2}

Substitute 5z+2=u

5\frac{dz}{du}=1

dz=\frac{1}{5}du

I_1=\frac{1}{5}\int\frac{du}{u}

I_1=\frac{1}{5}ln|u|+C_1

I_1=\frac{1}{5}ln|5z+2|+C_1

Now, solve RHS,

I_2=\int e^{\sin t}\cos t dt

Substitute \sin t=v,

\cos t\frac{dt}{dv}=1

\cos tdt=dv

I_2=\int e^{v}dv

I_2=e^{v}+C_2

I_2=e^{\sin t}+C_2

Subtitle the values of I₁ and I₂ in equation (1).

\frac{1}{5}ln|5z+2|+C_1=e^{\sin t}+C_2

\frac{1}{5}ln|5z+2|=e^{\sin t}+C_2-C_1

\frac{1}{5}ln|5z+2|=e^{\sin t}+C

Therefore the general solution of given differential equation is \frac{1}{5}ln|5z+2|=e^{\sin t}+C.

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