DSolveValue

Solves a linear, constant-coefficient ODE of order 1 or 2 for Y(x), with optional initial conditions.

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DSolveValue(eqn, Y(x), x)the general solution, with arbitrary constants C(1) (and C(2) for order 2).

DSolveValue({eqn, ic1, ic2, ...}, Y(x), x)the particular solution satisfying the stated initial conditions.

Typed signatures

Type	Package	Applies
(expression, expression, symbol) -> expression	analytic	always

Domain: Analytic

Details

• Y'/Y'' are written the way compute-engine itself represents an undifferentiated function's derivative -- D(Y(x), x) / D(Y(x), x, x) (boxing to Apply(Derivative(Y, n), x)) -- no new notation introduced. An initial condition is the same shape applied at a NUMBER instead of x: Equal(D(Y(x), x), b) at x=0 is Equal(Apply(Derivative(Y, 1), 0), b) for $Y'(0)=b$. Arbitrary constants print as C(1)/C(2), matching Wolfram's own C[1]/C[2] (confirmed directly: compute-engine's native C, called C(1), stays a plain, inert expression, exactly like Wolfram's own C[1]).
• Homogeneous part: the characteristic polynomial's roots -- a single real root (order 1); two distinct real roots; a repeated real root ($(C(1)+C(2)x)e^{rx}$); or a complex-conjugate pair ($e^{px}(C(1)\cos(qx)+C(2)\sin(qx))$).
• Nonhomogeneous part, by undetermined coefficients, ONE forcing shape at a time (not sums of different shapes): a polynomial in x (any degree), $Ae^{cx}$, or $A\sin(wx)+B\cos(wx)$. Each gets $x^s$ times its own trial basis, s = how many times the forcing's own characteristic value already appears as a homogeneous root (the standard resonance bump) -- confirmed against wolframscript's own DSolveValue for every case below, matching up to relabeling which root gets C(1) vs C(2).
• Every solution is verified before being returned: substituted back with D, the residual must come back exactly 0 symbolically or measure below 1e-9 at several numeric points -- declining, not guessing, otherwise.
• Declined: order > 2; a non-constant or non-numeric-literal coefficient (including a leading coefficient of 0); a nonlinear term ($Y(x)^2$, $\sin(Y(x))$, ...); forcing that mixes shapes or isn't one of the three above; and initial conditions at anything but a plain real-number point.
• Examples use the unknown function Y (capitalized) rather than y deliberately -- an unrelated repo-wide provenance check reuses one shared compute-engine instance across every reference example, and boxing Derivative(y, n) for a lowercase y marks that SYMBOL incompatibly typed there ever after, which then breaks other examples elsewhere that use bare y as an ordinary variable (confirmed directly). Y avoids the collision; the head itself takes any function name.

Examples

Basic #

$Y''+Y=0 \Rightarrow C(1)\cos x + C(2)\sin x$ — complex-conjugate roots $\pm i$

live ↗

In$\mathrm{DSolveValue}(Y^{\prime\prime}(x)+Y(x)=0, Y(x), x)$Out$\sin(x)C(2)+\cos(x)C(1)$

$Y''+Y=0,\ Y(0)=1,\ Y'(0)=0 \Rightarrow \cos x$

live ↗

In$\mathrm{DSolveValue}(\begin{cases}Y^{\prime\prime}(x)+Y(x)=0\\Y(0)=1\\Y^{\prime}(0)=0\end{cases}, Y(x), x)$Out$\cos(x)$

Scope #

$Y''-3Y'+2Y=0 \Rightarrow$ roots $1,2$

live ↗

In$\mathrm{DSolveValue}(Y^{\prime\prime}(x)-3Y^{\prime}(x)+2Y(x)=0, Y(x), x)$Out$C(1)\exp(2x)+C(2)e^{x}$

$Y''-4Y'+4Y=0 \Rightarrow$ repeated root $2$

live ↗

In$\mathrm{DSolveValue}(Y^{\prime\prime}(x)-4Y^{\prime}(x)+4Y(x)=0, Y(x), x)$Out$(xC(2)+C(1))\exp(2x)$

resonance — $Y''+Y=\sin x$, forcing frequency matches the homogeneous roots $\pm i$

live ↗

In$\mathrm{DSolveValue}(Y^{\prime\prime}(x)+Y(x)=\sin(x), Y(x), x)$Out$-\frac{x\cos(x)}{2}+\sin(x)C(2)+\cos(x)C(1)$

About

Examples	5
Checked against	Wolfram Language 4 agree1 unevaluated
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/dsolve.ts

See also: SeriesCoefficient, CaputoD