\(y=ae^{3x}+be^{-2x}\)...(1)
Differentiating both sides with respect to x, we get:
\(y'=3ae^{3x}-2be^{-2x}\)...(2)
Again, differentiating both sides with respect to x, we get:
\(y''=9ae^{3x}+4be^{-2x}...(3)\)
Now, multiplying equation(1)with equation(2)and adding to equation(2), we get:
\((2ae^{3x}+2be^{-2x})+(3ae^{3x}-2be^{-2x})=2y+y'\)
\(\Rightarrow 5ae^{3x}=2y+y'\)
\(\Rightarrow ae^{3x}=\frac{2y+y'}{5}\)
Now, multiplying equation(1)with equation(3)and subtracting equation(2)from it, we get:
(3ae3x+3be-2x)-(3ae3x-2be-2x)=3y-y'
\(\Rightarrow\) 5be-2x=3y-y'
\(\Rightarrow -2x=\frac{3y-y'}{5}\)
Substituting the values of ae3x and be-2x in equation(3),we get:
\(y''=9.\frac{(2y+y')}{5}+4\frac{(3y-y')}{5}\)
\(\Rightarrow y'' =\frac{18y+9y'}{5}+\frac{12y-4y'}{5}\)
\(\Rightarrow y''=\frac{30y+5y}{5}\)
\(\Rightarrow\) y''=6y+y'
\(\Rightarrow\) y''-y'-6y=0
This is the required differential equation of the given curve.
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A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
Read More: Formation of a Differential Equation