Question

If 𝒚_𝟏 = 𝒆^𝒙 and 𝒚𝟐 = 𝒆^−𝒙 are solutions of homogeneous linear differential equation, then necessarily 𝒚 = −𝟓𝒆^−𝒙 + 𝟏𝟎𝒆^𝒙 is also a solution of the DE

Short answer

Yes 𝑦 = −5𝑒^−𝑥 + 10𝑒^𝑥 is also a solution of the DE.

Step-by-step solution

Define First Order Differential Equation

A homogeneous linear differential equation is a differential equation in which every term is of the form 𝑦(𝑛)𝑝(𝑥)𝑦^∧{(𝑛)} 𝑝(𝑥)𝑦(𝑛)𝑝(𝑥) i.e., a derivative of 𝑦 times a function of 𝑥.

Solve the differential equations.

Yes, if 𝑦₁= 𝑒^𝑥 and 𝑦₂ = 𝑒^−𝑥 are solutions of a homogeneous linear differential equation, 𝑦 = −5𝑒^−𝑥 + 10𝑒^𝑥 is also a solution for this differential equation in the form 𝑦 = 𝑐₁𝑦₁ + 𝑐₂𝑦₂, since we have 𝑐₁ = 10 and 𝑐₂ = −5.

So, it is True that 𝑦 = −5𝑒 ^−𝑥 + 10𝑒^𝑥 is also a solution of the DE.