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UPSC 2017 Maths Optional Paper 1 Q6a-i — Step-by-Step Solution

8 marks · Section B

Particular integral via operator method · ODEs · asked 7× in 13 yrs · Read the full method →

Question

Solve the following simultaneous linear differential equations: $(D+1)y=z+e^x$ and $(D+1)z=y+e^x$ where $y$ and $z$ are functions of independent variable $x$ and $D\equiv\dfrac{d}{dx}$.

Technique

Symmetric system — decouple by forming $y+z$ and $y-z$, solve each first-order equation, recombine.

Solution

Step 1 — Add and subtract to decouple

Let $u=y+z$ and $w=y-z$. Adding the two equations:

Subtracting (first minus second):

Step 2 — Solve the two decoupled equations

For $u$: $\dfrac{du}{dx}=2e^x\Rightarrow u=2e^x+2C_1$ (write the constant as $2C_1$ for convenience).

For $w$: $\dfrac{dw}{dx}=-2w\Rightarrow w=2C_2e^{-2x}$.

Step 3 — Recover $y$ and $z$

Answer