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UPSC 2015 Maths Optional Paper 1 Q7a-ii — Step-by-Step Solution

6 marks · Section B

Laplace transform applied to IVP for second-order linear ODE with constant coefficients · ODEs · asked 10× in 13 yrs · Read the full method →

Question

Using Laplace transform, solve $y''+y=t,\;y(0)=1,\;y'(0)=-2.$

Technique

Standard Laplace solution of linear ODE with constant coefficients; partial fractions; inverse transform via known pairs.

Solution

Step 1 — Take Laplace transform

Let $Y(s)=\mathcal L\{y(t)\}$. Then

• $\mathcal L\{y''\}=s^2 Y-sy(0)-y'(0)=s^2 Y-s+2$.
• $\mathcal L\{y\}=Y$.
• $\mathcal L\{t\}=1/s^2$.

Equation:

Step 2 — Partial fractions on the first term

Multiply through: $1=As(s^2+1)+B(s^2+1)+(Cs+D)s^2$.

Expand: $1=As^3+As+Bs^2+B+Cs^3+Ds^2=(A+C)s^3+(B+D)s^2+As+B$.

Match coefficients:

• $s^3$: $A+C=0$.
• $s^2$: $B+D=0$.
• $s^1$: $A=0$.
• $s^0$: $B=1$.

So $A=0,\;C=0,\;B=1,\;D=-1$:

Step 3 — Substitute back

Step 4 — Invert

Answer