Related Formula
A first-order linear differential equation in standard form (dx)/(dt) + P(t)x = Q(t)$\frac{dx}{dt} + P(t)x = Q(t)$ is solved using the Integrating Factor:
I.F. = e∫ P(t) dt$$\text{I.F.} = e^{\int P(t) \, dt}$$
Core Logic
Let's rearrange the given differential equation into standard linear form:
(t+1)dx = (2x + (t+1)⁴)dt$$(t+1)dx = (2x + (t+1)^4)dt$$
(dx)/(dt) = (2x + (t+1)⁴)/(t+1)$$\frac{dx}{dt} = \frac{2x + (t+1)^4}{t+1}$$
(dx)/(dt) - (2)/(t+1)x = (t+1)³$$\frac{dx}{dt} - \frac{2}{t+1}x = (t+1)^3$$
Step 1: Compute Integrating Factor and General Solution
Here, P(t) = -(2)/(t+1)$P(t) = -\frac{2}{t+1}$ and Q(t) = (t+1)³$Q(t) = (t+1)^3$.
I.F. = e∫ -(2)/(t+1) dt = e-2ln(t+1) = (1)/((t+1)²)$$\text{I.F.} = e^{\int -\frac{2}{t+1} \, dt} = e{-2\ln(t+1)} = \frac{1}{(t+1)^2}$$
The general solution is given by:
x · I.F. = ∫ Q(t) · I.F. dt + C$$x \cdot \text{I.F.} = \int Q(t) \cdot \text{I.F.} \, dt + C$$
(x)/((t+1)²) = ∫ (t+1)³ · (1)/((t+1)²) dt + C$$\frac{x}{(t+1)^2} = \int (t+1)^3 \cdot \frac{1}{(t+1)^2} \, dt + C$$
(x)/((t+1)²) = ∫ (t+1) dt + C = ((t+1)²)/(2) + C$$\frac{x}{(t+1)^2} = \int (t+1) \, dt + C = \frac{(t+1)^2}{2} + C$$
Step 2: Apply Boundary Condition
Given the initial condition x(0) = 2$x(0) = 2$:
(2)/((0+1)²) = ((0+1)²)/(2) + C 2 = (1)/(2) + C C = (3)/(2)$$\frac{2}{(0+1)^2} = \frac{(0+1)^2}{2} + C \implies 2 = \frac{1}{2} + C \implies C = \frac{3}{2}$$
Hence, the specific solution curve is:
(x)/((t+1)²) = ((t+1)²)/(2) + (3)/(2)$$\frac{x}{(t+1)^2} = \frac{(t+1)^2}{2} + \frac{3}{2}$$
x(t) = ((t+1)⁴)/(2) + (3)/(2)(t+1)²$$x(t) = \frac{(t+1)^4}{2} + \frac{3}{2}(t+1)^2$$
Step 3: Evaluate at t = 1
Substituting t = 1$t = 1$:
x(1) = ((1+1)⁴)/(2) + (3)/(2)(1+1)² = (16)/(2) + (3)/(2)(4) = 8 + 6 = 14$$x(1) = \frac{(1+1)^4}{2} + \frac{3}{2}(1+1)^2 = \frac{16}{2} + \frac{3}{2}(4) = 8 + 6 = 14$$
Pattern Recognition
Sees: Linear form hidden under differential grouping coefficients.
Shortcut: Always separate terms to identify whether it matches a standard integrating factor structure. Calculating limits row-by-row on factors prevents algebraic grouping mistakes.
Chapter Mix
Class 12 Mathematics: Differential Equations
